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THE ROOTS OF TRIGONOMETRIC INTEGRALS
By N. G. pE Bruign

1. Introduction. Concerning the roots of trigonometric integrals G. Pdlya
(see references at the end of the paper) has proved a number of results which
he derived from properties of the roots of polynomials. He proved, for instance,
the reality of all the roots of the following functions:

(1.1) | ent et® dt . (n = 1,2,3,-- ‘);
(1.2) i] C(de* dt (> 0),
where C(f) = exp (—A cosh 4), and

(1.3) | exp (—at” + bt + cf’) exp tet dt,

where a > 0, b real, c > 0, n = 1, 2, 8, --- . (See concerning (1.1), [7], [8];

concerning (1.2), [6], [8]; concerning (1.3), [8].)
Another important result of Pdlya is.the following one (see [8]): Suppose
that the function F(é) of the real variable ¢ satisfies

F (8) integrable over —~ < t <o; F(t) = (F(—))*, -~< t <a;
(1.4)
F@® = 0€''") fort>£o, b> 2.

(The * indicates the conjugate imaginary.)
Let ¢(t) be an integral function of genus 0 or 1, with real roots only, and let
the number y be > 0. If the function F(d) is such that all the roots of the

integral
(1.5) | Fie dt

are real, then the same holds for the function f°. F(g(it)e’'e*”* dt.

The function ¢(if)e”’” is easily seen to be the limit of a sequence of poly-
nomials, all of whose roots are purely imaginary. Pédlya’s result, stated in other
words, is that these functions are universal factors, which conserve the reality
of the roots of any trigonometric integral whose integrand satisfies (1.4). Pdlya
also proved that the functions g(it)e”*’ indicated above are the only analytical
functions with this property. The latter result will not be used in the present

paper.
Received July 16, 1948.
197

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198 N. G. DE BRUIJN

In the sequel we continue Pélya’s researches. Our main results are

Turorem 1. Let f(t) be an integral function of t and such that its derivative
f'() is the limit (uniformly in any bounded domain of the t-plane) of a sequence
of polynomials, all of whose roots lie on the imaginary axis. Suppose furthermore
that f(t) is not a constant, and that f(t) = f(—t), f® = 0 for real values of t. Then
the integral [2.7 e'** dé has real roots only.

(The conditions (f() = f(—1), f® = 0) may be replaced by weaker ones,
namely, “f(é) = (f(—#)*, Ref® > Ofor —© <t<o”, if f(z) is a polynomial
or a function of the type (1.6) (see Theorems 19 and 20 respectively). It is not
easy to see whether the latter set of conditions is sufficient in the general case.)

Pélya’s results (1.1) and (1.2) are special cases of this one, but (1.3) is not.

Turorem 2. Let N be a positive integer and put
N
(1.6) PQ) = Dae’ (Re py > 0;p¥ = pn = 0,1, 2, +++).
—N
Let the function q(x) be regular in the sector —x/2N — N ~* arg py < arga <

a/2N — N™ arg py and on its boundary, with possible exception of « = 0 and
x = © which may be poles (of arbitrary finite order) for q(x). Furthermore suppose

(1.7) (q(z))* = g(1/2*)

in this sector (in other words, q(x) is real for |x| = 1. Then all but a finite number
of roots of the function

(1.8) a = [| ePQ(ne™ at (QW = a)
are real.

It may be remarked that our method fails to give any useful information
concerning the number and location of the non-real roots of (1.8) in the general
case, so that this very peculiar result may be of very little practical importance.

The special functions (C(é) = exp( —) cosh 2)

ol N
(1.9) we) = [ CO Dawe dt > 0,08 = an)

which have N pairs of non-real roots at most (Theorem 21), may be of some
interest’ since the Riemann £&function can be approximated by functions of
this type (see [8]). It will be worthwhile to determine classes of functions of
this type with the property that all the roots are real. We shall study these
questions in §6.

(Those readers who are mainly interested in considerations concerning the
éfunction may omit the proof of Theorem 1 and the related results in §5, and
in Theorem 2 need to consider the case P(t) = A(e’ + e7*) only. In that case
the complicated §4 is superfluous since the results of that section then reduce
to well-known asymptotic formulas concerning the I'-function.)

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THE ROOTS OF TRIGONOMETRIC INTEGRALS 199

In §7 we expose what progress has been made in this paper in the direction
of the Riemann hypothesis, and also how small this progress is.

An outline of the proofs of Theorems 1 and 2 concludes our introduction.

Sections 2 and 8 will furnish functions S(é) which are special universal factors
in Pélya’s. sense but which have stronger properties than those stated above.
A function S(é) of the real variable ¢, satisfying S() = (S(—d)* will be called
a strong universal factor if it joins properties (a) and (8) below, for any function
F@) satisfying (1.4).

(x) If the roots of (1.5) lie in a strip |Im z| < A (A > 0), then those of
f2. F@®S(@e*** dt lie in a strip |Im z| < A; , where A, < A, A, independent
of F(d).

(8) If F( is such that, for any « > 0, all but a finite number of roots of
(1.5) lie in the strip | Im z| < «, then the function f°. F()S(®e**' dt has only
a finite number of non-real roots.

It will be evident from (@) that any strong universal factor is a universal
factor in Pélya’s sense.

A function S(d) of the type

(1.10) S(t) = >> ae™ (A > 0, a, = a*,)
-N

is a strong universal factor if all its roots lie on the imaginary axis. (Conversely,
if S(£) is a universal factor and if it is of the type (1.10), then its roots lie on the
imaginary axis. This follows from Pédlya’s result on universal factors.) This
result is obtained by generalizing a theorem of J. L. W. V. Jensen on the location
of the roots of the derivative of a polynomial with real coefficients (§2) and
applying it to integral functions (§3).

The functions e7‘*, y > 0, also turn out to have property (a), but it is doubtful
whether they have property (@).

The functions (1.8) will be shown to have but a finite number of roots outside
any strip | Im z| < « ¢ > 0. This will be carried out by proving asymptotic
formulas for ®(z), depending on the expansion (5.7). In that formula an
auxiliary function H(s) occurs which is a generalization of the T-function.
Asymptotic formulas for H(s) will be derived in §4.

Now let P(@) and Q() satisfy the conditions of Theorem 2; then also P()
and Q(f)/(e’ + 2 + e‘) satisfy these conditions. From what is said above it
is evident that the function f°. e?°Q@ (ei + 2 + e')'e* dé has but a
finite number of roots outside any strip | Im z| < « Now applying property
(8) with S() = e° + 2 + e* we obtain Theorem 2.

In order to sketch the proof of Theorem 1, let P(@) satisfy the conditions of
Theorem 2 and suppose that P’() has purely imaginary roots only. Let A be
the smallest number with the property that the roots of @,(z) = f°. ee" dt,
which has but a finite number of non-real roots by virtue of Theorem 2, lie in
the strip | Im z| < A, and suppose A > 0. The function P’(é) is a strong uni-
versal factor and hence, by (a), the roots of f°. e? iP’ (He*” dt lie in a strip


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200 N. G. DE BRUIJN

|Imz| < A,, A, < A. But it is easily seen from partial integration that the
latter integral equals —z,(z). It follows that the roots of ®,(z) also lie in the
strip |Im z| < A, . This contradicts the minimum property of A. Hence
A = 0 and all the roots of ®,(2) are real.

It will be relatively easy to extend this to the integrals of Theorem 1 on con-
sidering F(é as the limit of a sequence of functions P().

The following notations are used throughout the paper.

Re a and Im e denote the real and imaginary parts of a:a = Rea +7Ima;a
denotes the conjugate of a. If f(z) is a function of the complex variable z, then
f*(z) is defined by f*(z) = (f(z*))*. A polynomial or integral function f(z) is
called real if f(z) = f*(z), that is to say if f(z) is real for real values of z.

All the trigonometric integrals considered in this paper are real integral
functions of z.

*

2. Theorems on polynomials. We shall deal with linear combinations of the
type (2.3) for a given polynomial f(z) with real coefficients; the simplest case
isfi() = fle+a2 +f — 1). Several properties of the roots of f,(z) are known;
they all express in some way that the roots of f,(¢) lie closer to the real axis
than those of f(z).

1. The number of non-real roots of f,(z) does not exceed that of f(z). (This
is a special case of Poulain’s theorem. See [11; Abschn. VI, Aufg. 63].)

2. If the roots of f(z) lie in the strip | Im z| < 1, then f,(z) has real roots
only. Namely, | f(z + 2)| # |f(@ — 2) | for Im z # 0. (Properties 1 and 2
are contained as special cases in Theorem 9a.)

3. If a, , «++ , a are the roots of f(z) and 8, , --- , 6, those of fi(z), then
>-* | Im g,| < DoT | Ima, |. (See [1; Theroem 5].)

These properties are meant for illustration and will not be used in the present
paper. We shall now derive a new result of this type, Theorem 3, which forms
the base of our paper. It is a generalization of the second property above. We
first prove

Lemma 1. Putz = 2 + ty (x and y real), and f(z) = 2 + A’ where A > 0;
let \ be a positive number. If pis defined by w = (A — 2°)? (A > d) and »p = 0
(A <2), then we have |f2@+i)| > lf~@ — a» | ¥ (Ve? — wy > 0
and |fie+a)| <|fe@-— alee — wy <0.

Proof. We evaluate | f(z + i) |? — |f@ — a) P = | {e+ ey +npP t+
A’? — | {e+ iy —-dP + AP = 8r@’ + y? + — A’). The assertion
directly follows.

Now consider an arbitrary real polynomial f(z) of degree > 0. It can be
written in the form

2.1) fe) = AT (@ - 4)" + 43 TT @ - 89,


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THE ROOTS OF TRIGONOMETRIC INTEGRALS 201

where a; , b; real; A; > 0, A> 0. Again, let \ be a positive number. To any
A, exceeding \ we construct the circular region C; , defined by (x — a)? +
yz < Aj — d’; if A; < \ we take C; to be empty. By S = S(f) we denote the

sum of all C; and the real axis. We now show

'THrorEeM 3. If f(z) is of the type (2.1) and \ > 0, & a complex number ~ 0,
then all the roots of the polynomial

(2.2) Ef(e + a) + fle — a”)
(which has real coefficients) lie in S.

Proof. We suppose ¢ to lie in the upper half-plane and. outside S. Abbre-
viating (2.1) we write f(g) = A] g:(z) [] A,(2).  Trivially | 4,(¢ + 2d) | >
| hs(& — 2) |, and it follows from Lemma 1 that also | g:(¢ + 2d) | > | gi( — ®) |-
Hence | f(¢ + 7A) | > |f(¢ — 2A) |. If & lies in the lower half-plane and outside
S, then | f(¢ + 2d) | < |f(¢ — 7) |. In both cases we conclude that ¢ is not
a root of (2.2).

We remark that the limit case of Theorem 3 for \ — 0 leads to a well-known
theorem of J. L. W. V. Jensen on the roots of the derivative of a polynomial.
(See [3] and [10; Abschn. III, Aufg. 35].)

We want to iterate the result of Theorem 3. Therefore, we first define a set
Sy = Sy(f), N = 1, 2, --- , which is the sum of the real axis and the regions
Ciw,i=1,--+,n. If A; > AN? we take for C,y the region N(x — a)? +
y’ < A? — NX’, which is bounded by an ellipse; if A; < \N?, C.y is empty.

It is readily deduced from Theorem 3 that if the roots of the real polynomial
g(z) lie in Sy(f) then those of fg(z + iA) + Eo(e — 2d), & ¥ 0, lie in Sy.i(f).

Tuzormm 4. Suppose that all the roots of the polynomial g(u) = doo a,u*,
ay % 0, Ke on the unit circle |u| = 1, that f(z) 7s a real polynomial, and that
1 > 0. Then the roots of

(2.3) T-9(1) fle) = Yi agfle -+ (Qk — Nya

k=0

are contained in Sy(f). Here T represents a translation operator defined by
T"f(@ = fle + tu).

Proof. The function u~"yg(u’) can be written in the form

ue) =a I] (yu ++ tu’) (a #0, & #0).

By Theorem 3, the real polynomial (£,T” -+ £*T~*) f(z) has its roots in S,(f).
A second application shows that the roots of (&7* + &T (4&7 + &T
lie in S.(f), etc., so that the roots of T-e(T”)f (z) turn to lie in Sy(f).

Turorem 5. Let f(z) be a real polynomial whose roots lie in the strip | Im z| <
A, A > 0, and let o(u) satisfy the conditions stated in Theorem 4. Then the roots


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202 N. G. DE BRULIN

of the polynomial (2.3) satisfy |Im z| < {A? — NX} if A > dN}, Imz = 0
if A < AN}.

Proof. Follows directly from Theorem 4 and from the definition of Sy(f).

8. Application to integral functions. Strong universal factors. Let a real
integral function be given of the type

3.1 f@ = Aete* TL = e/pne”™,

where A is real and ¥ 0, m is a natural number, a is real, p, ~ 0,| Imp, | < A,
> |e, |? < and the roots p, and p¥ have the same multiplicity. It is possible
to construct a sequence of polynomials f(z), fo(z), --- , all having their roots
in the strip | Im z| < A, converging uniformly to f(z) in any bounded region.
Since the product (8.1) converges uniformly in any bounded region, it is ob-
viously sufficient to prove it for the functions e”* (a real), (1 — 2/p,)e””” if
p, is real and (1 — z/p,)(1 — 2/p*) exp (2/p, + 2/p*) if p, is not real. In the
latter case p,' -+ p*~* is real, and thus it only remains to be proved that e”
(a real) is the uniform limit of a sequence of polynomials with roots only in the
strip | Im z| < A. We have indeed e** = lim,... (1 + az/n)”, converging uni-
formly in any finite region.

Since also e7°*” (6 > 0) is the limit of a sequence of polynomials with real
roots, the same applies to the function e’’ f(z), b > 0, if f(z) satisfies the con-
ditions mentioned above.

Conversely, it seems probable that, if a sequence of real polynomials with
roots in the strip | Im z| < A converges, uniformly in any bounded region, to
an integral function, then this function will be of the type e’*' f(z), where
the genus of f(z) is either 0 or 1. (The corresponding problem for functions
with real roots was solved by Pélya [4].) We do not need the solution of this
problem for our present purposes. (After this paper was written the conjecture
stated above has been proved by Mr. J. Korevaar.) Namely, we are able to
restrict ourselves to integral functions of order < 2; these functions satisfy
| f(z) | < exp (| 2 |"), » < 2, for | 2 | sufficiently large. According to Hadamard’s
theory such a function can be expanded into a product of the type (3.1) with
> | 0, |? <. If we now suppose that the roots of f(z) lie in the strip | Imz| <
A and that f(z) is real for real z, it immediately follows that f(z) has all the prop-
erties mentioned in the beginning of this section’ We thus obtain

Turorem 6. Jf the order of the real integral function f(z) is < 2 and if the
roots of f(z) lie in the strip |Imz|< A, A > 0, then there exists a sequence of real
polynomials f,(z) whose roots lie also in that strip, such that f(z) > f(z) uniformly
in any finite region.

For convenience we explicitly formulate the following well-known result (see
[10; 128, Abschn. 3, Aufg. 201).


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THE ROOTS OF TRIGONOMETRIC INTEGRALS 203

Turorem 7. If f(z), fi(z), fo(z), -+- are integral functions, f(z) not identically 0,
with f,(z) —> f(z) uniformly in any finite region, and if the roots of f,(2), fo(z), +**
all belong to a given closed point-set 8, then the roots of f(z) also lie in S.

Theorems 4 and 5 can now be applied to sequences of polynomials.

Turorem 8. If f(z) satisfies the conditions of Theorem 6 and ¢(u) those of
Theorem 4, then the roots of

(3.2) T™™ oT YQ O> 0, MI =fe + i)
satisfy |Imz| < (A? — Nd’) if A > AN}, Imz = 070 < A < AN}.
Proof. Let f,(z) — f(2) according to Theorem 6. It is easily seen that

(3.3) PMI), = TM o( PYF

uniformly in any finite region. By Theorem 5, the polynomials on the left have
their roots in the strip |Imz| = « = {Max (A? — NX’, 0)}*. Now the desired

result follows from Theorem 7.

TuroreM 9. Let the real integral function f(z) be of order < 2 and suppose that
f(2 has but a finite number of roots outside the strip |Imz| < A. If furthermore
g(u) satisfies the conditions of Theorem 4, then all but a finite number of roots of
(3:2) satisfy |Imz| < {Max (A® — Nd’, 0)}?.

Proof. We put f(z) = g(2)h(z), where g(z) is a polynomial, and the roots of
the integral function h(z) lie in the strip | Im z| < A. It is easily seen from the
arguments used in the beginning of this section that f is the limit of a sequence
of polynomials of the type f,(z) = g(z)h,(z), where h,(z) has no roots outside
|Im z| < A. We may actually take for h,(z) polynomials which have, apart
from a number of real ones, only roots which are roots of h(z) also.

According to Theorem 4, the roots of T~™*(T”)f,(z) lie in Sy(f,). It fol-
lows from the definition of Sy that Sy(f,) = Sw(he) + Sw(g) G R + Sy(g),
where R represents the set | Im z| < {Max (A” — Nn’, 0)}*. Now (3.3) and
Theorem 7 show that the roots of (3.2) belong to R + Sy(g). Sw(g) consists
of a finite number of ellipses. Since each ellipse contains ‘but. a finite number
of roots of (3.2), our proof is completed.

In the special case N = 1, \ > A we can obtain more complete information
on the number of non-real roots.

THEOREM 9a. If the real integral function f(z) of order < 2 has exactly 2k roots
outside the strip | Im z| < A, and if dX > A > O, — # 0, then the function
filz) = ef(2 + or) + Efe — 2) has 2k non-real roots at most.

Proof. The function W(z) = éf(¢ + 7A) has not more than & roots in the
lower half-plane. The theorem now follows by Lemma 2, §6.

In the general case \ < A of Theorem 9 the analogue of Theorem 9a is not
true. Even the following statement is false: if f(z) is a real polynomial with


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204 N. G. DE BRUIJN

2k roots outside the strip | Im z| < 1, then f(z), 0 < A < 1, has at most 2k
roots outside the same strip. Taking & = i, \ — 0, we should infer that f’(z)
has at most 2k roots outside that strip. This is incorrect, for instance, for
f@ = @+HE + 12), f'@ = Gee + 302)".

In order to be able to apply the preceding results to trigonometric integrals
we first state

Turorem 10. Let b be a number > 2, and lat the real or complex function F(t)
be integrable over —0© <t <@ and satisfy

(3.4) F(t) = (F(—9)* for all real values of t,

(3.5) F(® = Oo") (t+ &).

Then the trigonometric integral

(3.6) 1@ = | ” Fle" dt

represents a real integral function of order < 2.

A simple proof can be found in Pélya [8].

Trrorem ll. Let F(t) satisfy the conditions of the preceding theorem and
suppose that the roots of the function S(t) = My ae’, af = a, , au ¥ 0,
d > 0, lie on the imaginary axis. Then we have: If the roots (all but a finite number
of the roots) of (3.6) lie in the strip | Im z| < A then the roots (all but a finite number
of the roots) of the real integral function

(3.7) [ ” FN) S(Ne* dt

lie in the strip |Im z| < {A” — 4M)’}? of A > AGM), and are real if A <
AGM)’.

Proof. Since the roots of S(é are purely imaginary, the roots of the poly-
nomial ¢(u) = >.“ a,u”™ lie on the unit circle |u| = 1; hence g(u) satisfies
the conditions of Theorem 4 (2M@ = N). Now our theorem immediately follows
from Theorem 8 (Theorem 9) and from the fact that T”* J. F@e** dt =
fe. F(e*‘e**’ dt, whence

co} 2 M
reer) | Rie dt = | FO YX ae dt.
-0© 2 —-M

A slightly better result can be obtained if S(t) contains factors ge 1b gee,
For instance, the function S(f) = ge’ + 2 + Ee gives rise to the strip
{Im z| < {A? — 207}, but S@ = ge + ee™ gives the strip | Im z |
{A? — }*}? which follows from Theorem 8 (Theorem 9) on taking ¢(u)
& + &u. By iteration, we obtain

iA


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THE ROOTS OF TRIGONOMETRIC INTEGRALS 205

Tueorem 12. If S(t) = []Y (eM! + ee“), where |&| = 1, % > 9,
k = 1, 2, +--+ , N, and the roots (all but a finite number of the roots) of (3.6) le
in the strip |Im z| < A, then the roots (all but a finite number of the roots) of
(3.7) lie in the strip |Im z| < {Max (A — OY %, 0)}?.

Theorem 11 proves the statements (a) and (8) made in the introduction con-
cerning strong universal factors. Although it will not be used in this paper,
we shall prove here that also the functions e**"’, ” > 0, have property (q).
We do not yet know whether they have or have not property (8).

TrEeorem 13. If F(2) satisfies the conditions of Theorem 10, and af all the roots
of (3.6) lie in the strip | Im z| < A, then all the roots of g(z) = Jr F(t)e?e'** dt
lie in the strip

(3.8) [Im z| < {Max (a? — 2’, 0)}#.

Proof. By Theorem 12, the roots of gy(2) = f%» F(é) (cosh \t/N)*"-e**' dt
lie in the strip (3.8). Owing to Theorem 7 it is now sufficient to prove that
gx(2) — g(z) uniformly in any finite region. Now this follows from (3.5) and
from the fact that for p’ > »” we have

(3.9) e* (cosh M/NYY > @ EEE

uniformly in -© <t<o. (3.9) results from the inequality cosh y < et,
—« <y <«, whence (cosh \i/N)”” < e?””. .

For completeness we mention the following theorem, a slight extension of
Pélya’s result on universal factors (see [8]) which deals with the case A = 0.

TuroreM 14. Let F(t) satisfy the conditions of Theorem 10 and suppose that
the roots of (3.6) lie in the strip |Imz| < A. Let (2) be a real integral function
of genus 0 or 1 (that is, a function of the type (3.1)), with real roots only. Then
the roots of

(3.10) [ : F(te(ite’’ dt

lie in the strip | Im z| < A also.

A proof can be given by introducing quite trivial modifications in Pélya’s
proof for the case A = 0. It is, however, also possible to deduce Theorem 14
from Theorem 12.

It is hardly necessary to say that the function ¢(it)e*"’, a > 0, satisfies as a
“universal factor’’ as well as (2), that is to say that the roots of

(3.11) / ” F(to(itete* di

lie in | Im z| < A for any F(é) satisfying the conditions of Theorem 14. But
the roots of (3.11) even lie in a narrower strip, which can be shown by applying
Theorem 13 to (3.10).


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206 N. G. DE BRUIIN

4. Application of the saddle-point method. An important part in our con-
siderations will be played by the function

(4.1) Hs) = fe? ut du,
0

where Re s > O and
(4.2) gu) =u aya P™ $ au" fos $F ay -
We are interested in the asymptotic behavior of H(s) for Re s > 0, | s | large;
the natural number N and the coefficients a, , --- , @y remaining constant.
The a’s need not be real. For a; = --- = ay = 0 we have H(s) = sI'(s), so
that asymptotic formulas for the T-function will appear as special cases.

In the sequel, positive constants a, , a , --+ will occur, chosen sufficiently

large to suit some special purpose. These numbers may depend on N, a ,
-,@y , but not on s.

_ The integrand of (4.1) has, for s large, just one saddle-point £ in the domain

| arg u| < a satisfying

(4.3) ig’() = 8.

Namely, putting s = 2”, £ = w” (w and z are positive for s and u positive), we
obtain the equation

wY + (N = DNTaw*? + (N — Q)N egw? + oe + NO ay-w = 2",

whence, for | w| > a, , the function 1/z can be expanded into a convergent

power series 2° = wt — (N — 1I)N aw” + --- .. Solving this equation

by the Birmann-Lagrange inversion formula we obtain w= et Bz? +

Bye? + +++ whence w™ = 2%(1 + Boe? + Be? $Y = ML tf met +
-2

yee? + +++), convergent for |z| > a. It follows that, for |s| > ag, the only
solutions of (4.3) are

(4.4) é= s(1+ ns + ys 4 + +++),

where s°/", 5°", .-- are derived from one and the same branch of the fune-

tion s'’". We shall restrict ourselves to the regions | args | < 37, | argu| <7,
and hence we only have to consider the case where s’’” is positive for s > 0.

We henceforth divide into two cases,

Case a: 42/9 < |args| < $a,

Case 8: |args| < 41/9,
and we put L = 4|£|? in Case a, L = 4] £|" in Case 8, M = |€|'", g =
¢ — L# in both cases. (The constant 47/9 is of course not essential. In Case
8 it may be replaced by any other number < 437; in Case a however, it cannot
be replaced by arbitrary small positive numbers.) Our integration contour for
the integral (4.1) will be

J. A straight line from 0 to gq.

Il. The straight lime wu = & + ye, —L < y <©@; we notice that | arg el <

ix -+ 8 for | s | sufficiently large.


===== tmp/pdfs/o0176-debruijn/page-12.txt =====
THE ROOTS OF TRIGONOMETRIC INTEGRALS 207

The major contribution to H(s) is furnished by the integral along IT passing
through the saddle-point & Its value is

(4.5) / eu du = # | eX dy,
@ -L

where K(u) = g(u) — slog uand u = & + yf’. We have

dK /dy = {g'(u) — swe = uw {ug'(u) — é9’®}
(4.6)

= wd [aug W} lay) dy = we | @tug’ a} /a) ay.
It is easily seen from (4.4) that, for|s| > a,, Res > 0,y => —L, we have
(zy ful > BlEL | @tug'} fd) — 1] <a le",

(4.8) | arg (u/é) | < 37/8.

Tt follows from (4.6), (4.7), (4.8) that, if |s| > a» , Re K(u) decreases from
y = -—Ltoy = —M. Furthermore, for y > M we have, again by (4.6), (4.7)
and (4.8), for |s| > a,

(4.9) Re (dK/dy) > 4 \&/u|y > ty/lt+y) >%.
It follows that

—M
(4.10) | [ eX dy | <4 é|? | exp (-K¢ — MP) |,
(4.11) | / eke ay | < 8 | exp (—K& + Me) |.
M
We now consider the interval —M < y < M. Since M = £°", we have

jéut — 1 + y&?| < ay’? |El*, | s| > ao, and by (4.6) and (4.7) we have

| dK /dy —y + y°€*| < ay (ye | + | y°E* |) (|s| > a).
Hence, for -M <y <M, .
(4.12) | Ku) — K® — ay + hs? | < anl(l YE" | + [ye ),

MM
| e*™ dy
—M

— pK) unde? 41,8274 y- y y )}

(4.13)

= 4 On) + OC") 404 or +++ .)k

= (2n)ie XO (1 + OE ™)}.


===== tmp/pdfs/o0176-debruijn/page-13.txt =====
208 N. G. DE BRUIJN

From (4.12) we easily infer that the right members of (4.10) and (4.11) are
Oe /%e*™), go that (4.5), (4.10), (4.11) and (4.18) give

(4.14) | eye du = (Qré)te *® { 1 + oe *)} .
In the second place we consider the integral
(4.15) | oa? dy
0

for which the Cases « and 6 have to be treated separately.

Case a. Since g = & — 4|£|'s and 41/9 < args < 47, it is easily seen, on
drawing a figure and carrying out some elementary trigonometric calculations,
that | arg q| > | arg (e*7*” — 467°) | — € > 51/9 for | s| large. It follows
that, if | s | > a. and if wu runs through the straight line from 0 to g, the maxi-
mum value of |e’ | is attained at u = g. The same being true for | u* |
(since Re s > 0), we obtain

a
| eu? du
0

Jt was noticed before that, if | s| > a, , Re K(u) decreases from u = g tou =
£ — Mt, so that Re K(q) > Re K(é ~ Me), whence the left side of (4.16) is
Oe “%e*™) (gee (4.12)).

Case 8. |args| < 42/9. We putu = @,0<t<1lqgq=e-3lerre.
Then we have (see (4.6))

(4.17) dK/dt = q{g'(u) — s/u} = t*{ug'(u) — &9'@},

and ug’(u) — é9'(é) = Si fug’(W}' du, {ug’(w)}’ = 1 + OW”). Hence as
and a,, can be chosen such that for |u| > a3, |s| > a4,

(4.18) ug(u) - HOH =-h+aE-—u (a) <sinr/20).

(4.16) <|ge™® |.

Now it is easily seen from a figure that for |s| > as , |arg ( — u)| <
|arg £| < 92/20. So it follows from (4.17) and (4.18) that, for |s| > a6, Re
K(u) decreases if u runs through the straight line segment from vu = Uw =
a139/|q | to wu = g, whence |e *® | < | e"*® | for w on that segment.

Let » be the maximum value of |e’ | on the remaining segment from 0 to
Up. If u lies between 0 and uw ,0 < t < a; |q\|"°, we have

[eo *™ | < B | ef (u/U)°e * | < L | en

| ek |.

Since the constant ai, = » | e’” | is independent of s, we obtain

gq
| eu? du
0

Just as in Case a, we infer that (4.15) is O( */"e *).

(4.19) < (1+ ais) | ge *® |.


===== tmp/pdfs/o0176-debruijn/page-14.txt =====
THE ROOTS OF TRIGONOMETRIC INTEGRALS 209

TuHEorEeM 15. If b is a positive constant and H(s) is given by (4.1), we have
(4.20) H(s) = (Qré)te PET + O(s-”*) i,

uniformly for Re s > —b,|s|—0. Here & = 8s + yg V™ 4 ys 7/% 4 eee
(absolutely convergent for s large) satisfies ég'() = s.

(We notice that for VN = 1 we obtain g’(u) = 1, § = s and then (4.20) becomes
the familiar Stirling formula for I'(s + 1).)

Proof. For Res > 0 the result follows from (4.14), (4.16), and (4.19). In
order to be able to consider values of s in the left half-plane, we continue H(s)
by the formula

(s+ DH(s) = His +) +a(N —- DN “AG +1-N%
(4.21)
+--+ tayiN “Hs +N”,

which can be found by partial integration: (s + 1)H(s) = fee? g'(u)-u?™* du.
It follows from (4.21) that H(s) can be continued over the whole plane but for
the points s = —1, —1 — 1/N, —1 — 2/N, --- , which are possible poles of
H(s). It also follows that (4.21) holds for all values of s except for these points.

The function A(s) = (2r8)'e @#, & = s + ys’ + --- , is regular and
satisfies

(4.22) s°h(s — p)/h(s) = 1 + O(s°-”%)

uniformly for 0 < p < 6+ 1, |args|<-7,|s| >a. This can be shown by
some elementary calculations. Now suppose that (4.20) is true for the region
Re s > —k/N. Then, by (4.21) and (4:22), it can be verified for Re s >
—(k + 1)/N. Hence the theorem follows by induction.

In the following Theorems 16 and 17 we shall collect some results concerning
H(s) to be used in the next section.

TororeM 16. Puts = o + tr (a andr real), and let b and ¢ be positive constants.
There are constants A and C (A and C may depend on a, , -*- , ay) such that,
for —-b <a <e|7|,|7| > A and any positive number p we have

(4.28) | H(s — p)|<C'? |s if" | Hs) |.

Proof. Take C = {1 + f[a,| (1 — 1/N) + =: + |aw-s |/N}QA + &)*
It follows from Theorem 15 and from (4.22), that A, A > C, can be determined
such that for 0 < p < 1,0 > —b,|7| > A, we have | s°H(s — p)/H(s)| <
(1 + ¢?)? < C"*’. Now the case p > 1 can be proved by induction. Suppose
that (4.28) is true for 0 < p < k/N (k an integer, k > N), and that k/N <
pi < (& + 1/N. Then we have | H(s — pi + 1)| < C” | H(s) |/js [°° ,
|H(s — p. + 1— 1/N)| < C7 *” | H(s) |/| s |?*7-7*", and so on, and hence,
since |r| > A > C,|H(s — p. +1 —&/N)| < CC" |s | | A(s) |, & = 1, 2,

-,N — 1. It now follows from (4.21) that


===== tmp/pdfs/o0176-debruijn/page-15.txt =====
210 ‘N. G. DE BRUIJN

|(s — pr: + 1)H(s — pi) |
< (+ fa: |(N — DN 4+ ++: + Jaw | NC" [8 |" | HO |.

Since « < c |r| and p, > 1, we have |s— a, + 1| > |s|1+ c’)* and
(4.23) follows for p = pi.

TuroreM 17. If b, c and 6 are given positive numbers, then positive numbers A
and C can be found such that | H(s) | > Ce Grr il¥e" gs = g + tr, in the region
o> —b,|s| >A. .

This follows from Theorem 15 by some simple calculations.

5. Proof of Theorems 1 and 2. In order to complete the proof for Theorem 2,
which was outlined in the introduction, it is sufficient to show (Theorem 18)
that the integral (1.8) has but a finite number of roots outside any strip |Imz|<
¢,¢ > 0. Theorem 2 follows (for Q(é)) by applying Theorem 18 to the function
Q(t) = Q@/(e’ + 2 + &') which also satisfies the conditions of Theorem 2,
and using Theorem 12 or 11.

We shall establish a series expansion (formula (5.7)) for (z) which generalizes
a formula of Polya (see [6; formula (11)]) for the function (1.2). The occurrence
of Q(t) in our S() makes it very difficult to carry out Pélya’s method in the
present case. We therefore develop a new method which uses contour inte-
gration.

Let the numbers a; , a2 , «** , ay be defined by pry + pray bee +
dn, = (py"y)™ + oa(py*y)* *? + -+- + ay and define g(u) by (4.2). Then
we have, by (1.6),

(5.1) P(t) = g(pye™') + g*(pie™").

Let B > 0 be such that Q(d is regular for | Ret| > B. Consider the following
paths W,, We, Ws in the complex plane.

“W, consists of a half line from 2ri + © up to the point 217 + B, the line
segment from 2ri + B to B, and the’real axis from B to +.

W, is the contour of a rectangle, taken in positive direction, with vertices B,

Qri + B, Qari — B, —B.
W, consists of the segments —» to —B, —B to 2x1 — B and 2x7 — B to

2ri — 2.
Considering e”“’ Q(e*** as our integrand, we immediately find that

For t on W, , and B sufficiently large, we have an expansion

(5.3) (exp (— goto ™ 00 = ae Be x)


===== tmp/pdfs/o0176-debruijn/page-16.txt =====
THE ROOTS OF TRIGONOMETRIC INTEGRALS 211

converging absolutely: and uniformly with respect. to t. Hence

(5.4) [ = > 8, [ (exp (<glpye))e"t* de

The integral

(5.5) | (exp (alone er" a

converges for any complex value of z. To evaluate it, first suppose Im z < 0,
then shift the vertical part of the path W, infinitely to the left. It follows that
(5.5) equals :

of (exp (—glpxe))e'* a

and hence
(5.6) | (exp (—g(pne™)))e"*' dt = NL — €?")py'* "HN — 1),
Wr ia

where H is the function introduced in (4.1). Since H(s) is regular over the
whole plane, with exception of simple poles at s = —k/N, k = 1, 2, --- , the
right side of (5.6) is an integral function, and (5.6) holds for all z.

We can deal with fy, in the same way, and after that (5.2) and (5.3) lead to
expansion

ve,

B@) = = apne 2 1)

(5.7) + vy pipe yo 1)
ao —PUO Ap _iet
+ 1—e?"? i: é Qe dt = ®,(2) + ®,(2) + ,(z).

This formula holds for all values of z, with the exception of the points z = ik,
k = 0, +1, £2,---.
We are now able to prove

THEOREM 18. If P(é) and QQ) satisfy the conditions of Theorem 2, and if we put

(5.8) g(2) = N'p_xpy°" "H(z + K)N* — 1),
then we have, for « > 0,
(5.9) @(z) = g(z){1 + o(1)}

uniformly in the half-plane Im 2 < —«, and ®(2) has but a finite number of roots
in that half-plane. Since (®(z))* = ®(z*), an analogous result holds for Im z > ¢.


===== tmp/pdfs/o0176-debruijn/page-17.txt =====
212 N..G. DE BRUIJN

Proof. We first consider the point-set & defined by the inequalities
(5.10) —|Rez| < Imz < —e (|z| > A).

(On generalizing Theorem 16 it is possible to extend our considerations to the
regions defined by the set of inequalities | Re z| > 1, Imz < —« j2| > A.
The power of the Phragmén-Lindeléf theorem applied below however enables
us to restrict ourselves to the smaller regions indicated above.) Here A is
chosen sufficiently large to suit some conditions indicated in the following.

By Theorem 16, the first series of (5.7) satisfies

(11) | &@) — e@ | S DV | BBR PMC |

< C2, )21°"" | © |,

if A is sufficiently large and ze R. (Ci, Co, --+ may depend on N, K, p; ,
8; , but not on z.) Analogously

(5.12) | B(2) — o*(—2) | < Cs |2[" | o(-2) |.

Furthermore we have, again by Theorem 16, taking » = (7 — 7z*) /N,
(e+ K _ )/= (4K _ ) anes (24% _ )|
H (=##* —1))=\9 1) <eler”| HEE 1

N
for ze R and A sufficiently large. Hence, by (5.11) and (5.12)
(5.18) (2) + &(2) = (2) {1 + o()}.

We now turn to the third part of (5.7), z.e., the function (2). It follows
from the regularity properties of Q() (see Theorem 2). that the path W. can be
reduced to the path W7, consisting of a rectangle with vertices B+ @r+6-
arg py)i/N; B + 20i + (—3m — 6, — arg py)i/N; —B + 2mi + (-99 — 6, -
arg py)i/N; — B + (ia + 4, — arg py)i/N for a certain positive number 6, .
The rectangle being independent of z, we find, for Im z < —«,

| (bey I.

C; exp {B | Imz| — (a + 4) | Rez/N | + arg py Rez/N}.

IA

From Theorem 17, with c > BN, 6 < 4, , we now easily derive that (see (5.8))
(5.14) ®,(z) = g(z)o(1) (|z| 72)

uniformly in the region considered.

It follows from (5.13) and (5.14) that (5.9) holds uniformly for ze &.

In the region R, defined by |Im z| < — | Re z|, |#| > A, we have, by
Theorem 10, &(z) = O(exp |z)), 4 < 2. On the boundary of R, we have


===== tmp/pdfs/o0176-debruijn/page-18.txt =====
THE ROOTS OF TRIGONOMETRIC INTEGRALS 213

&(z) = o(z){1 + o(z)} and furthermore | g(z) | > 1 for z in R, , provided that
A bas been chosen sufficiently large. It follows, by a well-known theorem of
Phragmén-Lindeléf, that (5.9) holds uniformly in R, .

Since H(s) has only a finite number of roots for Re s > —}, which follows
from Theorem 17, now Theorem 18 is completely proved. Hence Theorem 2
is true (see Introduction).

We now turn to the proof of Theorem 1. In the first place we deduce from
Theorem 2 and Theorem 11, by an argument explained in the introduction,

TrErorem 19. If P(t) = >oxy pe’ (N > 0, Re py > 0, pF = p-x) and if all
the roots of its derivative P’(t) are purely imaginary, then the function f°. e"e***
di has real roots only.

We easily infer

TuroreM 20. If all the roots of the derivative of the polynomial f(t) = Sy
dat’ (N positive and even) are purely imaginary, and if qv > 0 (so that f(t) is
real for t purely imaginary), then the function V(z) = J. e% e? dt has real
roots only. ,

Proof. We put f’(®) = Nav [LX (¢ — tp,), p, real, and for \ > 0,

N-1 é
Now []Asinh (= ip =a, | ale) dr = Wit.
vel
yy (é) has the form

N-1
(5.15) » ee.

—-N+1

Since N is even we have cy) = 0; hence ¥,(é) also has the form (5.15). It is easily
verified that ¥,() is real for purely imaginary values of ¢, and furthermore

Y(nd) = 2N(N _ 1)7* 4d) qye re P* + wee

It follows, that, for) > Ay = 2r* | Di ”'p, |, the function P,() = y(A%)
satisfies the conditions of Theorem 19, so that ®(z) = f%. ee’ dt has
real roots only. It remains to be proved that (see Theorem 7)
(5.16) lim ®(z) = ¥(z)

Ota

uniformly in any bounded domain of the z-plane.
We can determine positive constants 6, < 1/2N and 6, such that

(5.17) | (sinh w)/w | > 5, | arg (sinh w)/w | < 1/2N,
for any value of w satisfying | arg w| < 6, and |Im w| < 6,. Furthermore

we can fix positive numbers A, and A, such that the numbers w = (¢ — tp,)/A
lie in that region for all values of > A,,\’ > A, andvy,1<»<N—1. It


===== tmp/pdfs/o0176-debruijn/page-19.txt =====
214 LN. G. DE BRUIIN

now follows from (5.17) that a positive constant ¢ exists such that Re g,(f) >
ct’ fort > A,,\ > A.. Hence

(5.18) lim lim sup

Are per)

©
i e PDett at| = 0
A
uniformly in any bounded domain of the zplane. The same holds, of course,
for fz2.

Since gy > 0, N even, we also find that

A

t

(5.19) lim

Ave
uniformly in any bounded domain, and the same for f 74. For any fixed value
of A we have

A A
(5.20) lim ee dt ae / ef Pei dt
Ana VA —A

uniformly in any bounded domain, because ¥,(4) > f@® uniformly in —A <
ti<A.

From (5.18), (5.19) and (5.20) we infer (5.16) and our theorem is proved.

Proof of Theorem.1. Suppose that f(é satisfies the conditions of Theorem 1.
These conditions can also be expressed in the following form (see [4]): f’(@®
is of the type f’() = ae’ t*** [[?_. (1 + 834") where a > 0, b > 0, & an integer
> 0,6 >0,» = 1,2,°:-, DP s& <@. Now let f,@ be defined by fi@ =
a(1 + bf /n)"e*** T]? (1 + 3,4), f.(0) = f(0) and put B(z) = f2. 67"? ef dt,
®,(2) = f2.e 7 e** dt. Tt is easy to find positive numbers A, ¢ and 7 , such
that for n > m,t > A ort < —A we have f(t) > ef andf,() > et’.

Since f,() — f@® for n >, uniformly in any finite ¢interval, we now easily
infer that (see proof of Theorem 20)
(5.21) lim 4,(z2) = &()
uniformly in any bounded domain of the 2-plane.

It-is easily seen that the polynomials f,(z) satisfy the conditions of Theorem
20, whence it follows that #,(z) has real roots only. Application of (5.21) and
Theorem 7 completes the proof of Theorem 1.

6. Functions of the type (1.9). We consider the functions of the type (1.9).
Tn connection with the Riemann hypothesis (see §7) it may be important to
investigate classes of functions of this type.

In the following we shall prove some results concerning these functions.
Very little of the preceding sections will be needed here, since the reality of
the roots of f°. C(He” dt, C® = exp (—A cosh 2), (special case of Theorem 1
or 19) was already proved by Pélya (see (1.2)). Only Theorem 25 requires
asymptotic expressions and strong universal factors as its tools.

First we prove


===== tmp/pdfs/o0176-debruijn/page-20.txt =====
THE ROOTS OF TRIGONOMETRIC INTEGRALS 215

Lemma 2. If U(z) and V(z) are real polynomials such that W(z) = U(z) +
iV (z) has n roots in the lower half-plane, then U(2) has n pairs of conjugate complex
roots ai most.

(The case n = 0 is the well-known theorem of Hermite-Biehler (see [10;
Abschn. 3, Aufg. 25]).)

Proof. We may assume that U(z) and V(z) have no real roots in common,
so that W(z) has no real roots. We also assume that the degree m of W(z)
satisfies m > 2n, for otherwise the result is trivial. Now if 2 runs through the
real axis from — © to », the argument of W(z) increases by an amount of
a(m — 2n). Hence there are at least m -- 2n different points on the real axis
where W(z) is purely imaginary. If z = © is not one of these points, we thus
find m — 2n real roots of U(z) at least, and otherwise at least m — 2n — 1.
But in the last case the degree of U(z) must be < m, so that U(z) has at most
2n complex roots in both cases.

TurorEeM 21. The function (1.9) (which has but a finite number of non-real
roots on account of Theorem 2) has N pairs of conjugate complex roots at most.

Proof. The functions, where C(é) = exp (—) cosh #), A > 0,

(6.1) Oa ; Cel" at

62 &@ = | (CO cosh De“ dt, ®@) = [| (CWisinn eat

have real roots only (Theorems 1 and 11), whereas the roots of @(¢ — 7) =
®,(z) — 7®,(z) lie in the upper half-plane.

By partial integration it is easily seen that, for k = 0, 1, 2, --- , the functions
2’ ,(z) and z’@,(z) are of the form

k+1

23,2) = | “OSS aBerte!** dt (I = 1,2)

a n=—k-1
where a®@ = a , and the highest coefficients a4,,,,1 = 1, 2, are # 0. Further-
more ai") is real if & + lis odd, and purely imaginary if k + liseven. It follows
that the function Y(z), given by (1.9), can be expressed as a linear combination
of B(z), B(z), 2b:(2), «++ , 2” (2), &(z), --- , 2&2), with real coeffi-
cients. Since z6(z) = —A,(z) we have

(6.3) zU(z) = A@&(@) + BE) &),

where A(z) and B(z) are real polynomials of degree N at most.

Now if F(z) and G(z) are real polynomials, of arbitrary degree, with the
property that the roots of F(z) — <G(z) all lie in the upper half-plane, then
A(z)F(z) + B)G(z) has at most N pairs of conjugate complex roots. This
follows by application of Lemma 2 to the function U(z) + iV(z) = (A) +


===== tmp/pdfs/o0176-debruijn/page-21.txt =====
216 N. G. DE BRUIJN

1B(z)) (F(z) — iG(z)). On approximating ®(z¢ — 7) = (2) — 7®,(z) by poly-
nomials F(z — 7) whose roots also lie in the upper half-plane, we find that
(6.3) has N pairs of conjugate complex roots at most.

In connection with the Riemann hypothesis it appears to be important to
find large classes of functions

(6.4) Dy one . (og = an)

with the property that the integral (1.9) has real roots only. For instance, this
follows from Theorem 11, if all the roots of (6.4) lie on the imaginary axis; but
this result does not help much in the direction of the Riemann problem, which
seems to be related to functions of the type

2 N
(6.5) [ CO TT (a + cosh* pe a,

where pf, , -*- , p, are real and > 1. In the following we establish some new
results, which do not help either, but which may serve as material for obser-
vation.

Turormm 22. If > 0, u > 0, the integrals Wy(z) = J2. C()(u + cosh te’*
dt and W.(z) = S22 C(é)(u + cosh? te’*’ dt have real roots only.

Proof. (The reality of the roots of V,(z) can also be proved as follows. It
follows from Theorem 21 that V,(z) has at most one root in the upper half-
plane. Since ¥,(z)-is real and even, this possible one must be purely imaginary.
But since the integrand is positive for z¢ purely imaginary such a root does not
exist. The same argument is used in the proof of Theorem 23.) By partial
integration we easily express VW, and , in terms of the functions #,(z) and
®,(z) defined by (6.2):

(6.6) 2W,(2) = 28,@) — wo),

(6.7) he, (2) = 2&(z) + [2 — ML + »)}&@).

The roots of the polynomials z — iju and z + z{z’ — °(1 + y)} He in the
upper half-plane and hence, by the lemma, (6.6) and (6.7) have real roots only
(see the proof of Theorem 21). We notice that the same can be said if -1 <
p < 0, for then » + cosh ¢ and » ++ cosh”é are universal factors.

On the other hand it is not difficult to show that both integrals have a pair
of purely imaginary roots if — is positive and sufficiently large.

The following theorem is obtained by a generalization of the method employed
above. The result, however, seems to be too complicated for application to any
wide class of polynomials.

THEOREM 23. Let be > 0 and let f(y) bea real polynomial, such that &(z) =
Jew C(t f (cosh fe** dt has real roots only. Then the same holds for the polynomial


===== tmp/pdfs/o0176-debruijn/page-22.txt =====
THE ROOTS OF TRIGONOMETRIC INTEGRALS 217

fy) = wyfy) +FY) +N GY) FNPF Y) + +++ fu > 0. It is also true for
yu <0, provided that f,(y) does not change sign for y > 1.

Proof. Putting

a) = / (C( flcosh #) cosh 8) e*** dt,

a) = | ; (C(t) f(cosh ti sinh 2) e'** dt,

we find by partial integration
(6.8) 2 | C( f,(cosh fet! dt = ne®r(2) — E70).

Furthermore, we notice that the roots of ®’(z — 7) = @7(z) — 77 (2) all have
imaginary part 1. For » > 0 the function A(z) + iB(z2) = pz — a has no
roots in the lower half-plane so that the argument used in the proof of the pre-
ceding theorem shows that the function (6.8) has real roots only.

If u < 0 we infer that (6.8) has one pair of conjugate complex roots at most.
Since that function is an odd function of z, these possible roots must be purely
imaginary. But if f;(y) does not change sign for y > 1 and if z is purely imagi-
nary, then C(é)f,(cosh é)e**’ does not change sign for ~~ <t<o. It follows
that the integral does not vanish.

The following application may be of some interest. The function f(y) = y
satisfies the conditions of Theorem 23, for (6.1) has real roots only and y* =
(cosh 2)” is a universal factor. Consequently Theorem 23 shows that, for \ > 0,

oo 2 2 N N-
(6.9) | (ol + cosh ¢ + Aco fovee ob MT east the dt

N

has real roots only.

TueorEeM 24. Let the polynomial f(y) of degree N have negative roots only, and
let d be a number > 4N. Then the function B~(z) has real roots only.

It may be surmised that the condition \ > 3N can be replaced by a much
weaker one.

Proof. We may assume that f(y) has no roots for —1 < y <1, the factors
y — a= cosht — a (—1 < a < 1) being universal factors (Theorem 11). We
now proceed by double induction, in the first place with respect to the degree
N of f(y) (the theorem is true for N = 0) and secondly with respect to a number
n, the smallest positive integer with the property that f(y) has at least one root
exceeding —1 — »/d. We shall reduce the case (N, 1) either to lower N or to
lower 7; the case (NV, 0) will always be reduced to lower N.

Suppose that f(y) (of degree N) has negative roots < —1 only, the largest
of which, p say, satisfies -—1 — n/X < p< -1— (n-—1)/,n> 1. Now


===== tmp/pdfs/o0176-debruijn/page-23.txt =====
218 N. G. DE BRUIJN

consider the polynomial f;(y) = f’(y) — Af(y). Since f(y) has real roots only,
the same applies to fi(y). Now f,(y) has no roots for y > 1 and exactly one
root p, satisfying p + ”* <p, < 1. Namely, f’(y)/f(y) decreases monotonically
for y > p. For y = p + d* we have f’(y)/f(y) > (y — op)’ = A, and, since
the roots of f(y) are supposed to be < —1, the inequality f’(1)/f(@) <3N <y
holds.

Now if p; < —1, the polynomial f,(y) belongs to a case of lower n and hence

(6.10) [ ” C(t) f(cosh De" di

is supposed to have real roots only. If, however, —1 < p, < 1, the polynomial
foly) = f.(y)/(y — p.) is of degree N — 1, and it follows that the roots of (6.10),
with f, instead of f, , are real. But since in that case cosh ¢ — p; is a universal
factor the same applies to (6.10) itself.

Again applying a universal factor, we find that

if C(é) f,(cosh t) sinh te**’ dt = z | C(i) f(cosh te** dt

has real roots only. This completes our induction.

We have not yet been able to generalize Theorem 24 to cases where f(cosh #)
is an infinite product of factors 1 + c, cosh é, ¢, > 0. Such an extension might
be given perhaps by carrying out a suitable reduction process and using Theorem
25 below. In Theorem.26 we shall use such a method in a different case, where
a reduction process can be found indeed. Theorem 26 may be of some interest
since it gives a result of a type we should like to have for functions of the form
(6.5).

Turorem 25. If > 0,0 < 6 < 4a, and if n is a natural number, then there
exisis a positive number A(A, 6, n) with the property that the roots of
(6.11) we) = | Cl sfeosh de a
lie in the strip |Imz| < AQ, 6, n) for any real polynomial f(y) of degree n whose
roots le in the sector 4a + 6 < arg y < 3a — 6.

(It is possible to prove the same for the region consisting of the real axis and
a circle |z| < AQ, 6, n). This can be done by introducing a denominator
(e’ + 2 ++ e”‘) (see the proof of Theorem 2).)

Proof. The function (6.1) can be developed as follows:

(-1)’ T@ — ») (—1)’ I(—zz — v)
P@) = are yt (RN) P+ (ay

(see (5.7) or [6]). It follows that for Im z < —1 we have
| ()(2a)**/T@z) — 1] < [e[°AQ),


===== tmp/pdfs/o0176-debruijn/page-24.txt =====
THE ROOTS OF TRIGONOMETRIC INTEGRALS 219

A = A(A) independent of z. Consequently, the function

d

@,(2) = [ " C(6)(cosh #)*e'*! dt = 27 > (H@{z + (ke — Qu)s}
satisfies
(6.12) | B(2)a****2-** /T (az) (iz)® — 1| < AQ, B)/l2| (Im z < —1).

Let A,(A, 2) be the largest of the numbers A(A), A(A, 1), --- , AQ, 2), and
let fy) = A+ ay) +--+ ey) = a + ay +--+ + 4,y", where | arge,| <
gx — 6, a, real. Then we have ¥(z) = a &(z) + a:4,(z) + +++ + a,8,(2),
and by (6.12), for Imz < —1,

Roca) — >> a,(iz/d)* | < Ap(r, n) | 2 [7 > | az” | x7,

Consequently
(6.13) TD I] 1+)|< iz] IT 1+) )-

Here ¢,z/h = d, satisfies | arg d, | << * ~ dandso|1+d,| > sin $6 (1 + | d, 1)
is easily verified. It now follows from (6.13) that W(z) ~ 0 if 2 satisfies Im z
< —landjz| > A,(, 2) (sin $8)~”. Since W(z) is real for real values of z, an
analogous result holds for z in the upper half-plane, so that A(A, 6, n) = Max
{A,(A, n) (sin 36)”, 1} has the required property. —

THEoREM 26, Suppose that \ > 0 and that the roots of the real polynomial f(y)
all lie on the negative real axis. If f’(1)/f() < 2a, then the function

(6.14) vw = [ ” €,(t) f(cosh te! dt,

where C(t) = exp (—X cosh’ 2), has real roots only.

Proof. We use the following Lemma which is not actually a special case of
Theorem 25 but which can be proved in the same way without any essential
difficulty. (First transform ¢ = 37.) In fact it is an easier case; we omit its
proof.

Leva 38. If > 0 and n a natural number, then there exists a positive number
AQ, 2), such that the roots of (6.14) satisfy | Im z| < AQ, 2) for any polynomial
Sy) of degree n with negative roots only.

For given values of \ and n, let Ao(A, 2) be the smallest possible of the numbers
A with the following property: The roots of (6.14) lie in the strip |Imz| < A
for any polynomial f(y), of degree < n, satisfying the conditions of our theorem.
Lemma 3 shows that such a number A,(A, 7) exists. We shall assume A)(A, n) >
0, and show that to be contradictory.


===== tmp/pdfs/o0176-debruijn/page-25.txt =====
220 N. G. DE BRUIJN

Let the polynomial f(y) of degree n satisfy the conditions of our theorem.
By partial integration we find -

(6.15) ave) = | ” (C.() fAcosh Di sinh 2) e* di,

where f,(y) = f"(y) — 2duf().
Since f’(1)/f() < 2A, f:fy) has exactly one root, say a, in the interval

0<y< 1. Now the polynomial fely) = fi(y)/Y — @) has n negative roots
Pr» P2, 7°" > Pn» satisfying

where o; , --- , ¢, denote the roots of f(y). This is easily seen by drawing graphs
of fi (~/fY and of 2dy. It follows from (6.16) that f2()/f2(1) < f’D/FM),
so that f.(y) also satisfies the conditions of Theorem 26. Hence the roots of
f°. Co(t)f2(He'** dé lie in the strip |Im z| < Ad(A, n).

Since i sinh ¢ and (cosh ¢ — a) are strong universal factors the roots of (6.15)
lie in a narrower strip |Im z| < AQ, 7), AiQ, 2) < AdQ, m). Namely, by
virtue of Theorem 11 we may take A,(A, n) = {Ag(A, 2) — 1}? if A, > 1 and
AA, 2) = Oif Ay <1.

_ As this holds for any polynomial f(y) satisfying the conditions of our theorem
it contradicts the minimum property of Ao (x, n). It follows that A.(A, n) = 0,
which proves our theorem.

The condition f’(1)/f(1) < 2\ in Theorem 26 isa natural one. It is equivalent
to the condition that the function C2(é)f(cosh £), occurring in the integrand of
(6.14), has its maximum for ¢- = 0 and decreases steadily for t > 0 (it is, of
course, a real and even function of ¢).

Theorem 26 can be extended to infinite products immediately.

THEOREM 27. If c:,¢:, +++ are positive numbers and if dee (1 + e,)7* < 2d,
then the function J. C2(6) []z-. (1 + ¢, cosh ée** dt has real roots only.

7. Remarks concerning the Riemann hypothesis. The special interest devoted
to integrals of the type (1.9) arose from the Riemann hypothesis concerning the
¢-function. Putting s = $+ 7z and writing (see [12; Chapter 3])

(7.1) Ee) = &s) = 4s(s — DT Gsr sO),
(7.2) ol) = Do Onbate* — Bntac NER",
we have

(7.3) x2) = | ; olde" dt;

here ¢(é) is an even function of ¢. Riemann conjectured that (7.3) has real roots
only.


===== tmp/pdfs/o0176-debruijn/page-26.txt =====
THE ROOTS OF TRIGONOMETRIC INTEGRALS 221

It is known from Euler’s product expansion ¢(s) = [], (1 — p™*)7* and from
the functional equation Z(z) = =Z(—z) that the roots of (7.3) lie in the strip
| Im 2z| < $ anyhow. Hence it follows, by Theorem 12, that

©

(74) i v(t) cosh $4 e!** dt

has real roots only. (See [9].)

It would be very interesting to have a proof for the reality of the roots of
(7.4) which does not use the known fact that ¢(s) # 0 for Res > 1, but only
employs the properties of ¢(é) cosh 3¢ and the general properties of trigometric
integrals. Such a proof might open the way to the Riemann hypothesis.

Neither Pélya’s work nor the present paper gave any direct information con-
cerning these problems thus far. But there are some interesting facts. Pélya
[8] noted that, iff — 1,

(7.5) e() ~ ¢,(t) = 42° (cosh 9¢/4) exp (—2z cosh £)

where [°. ¢,()e’*' dt has real roots only, since cosh 9é/4 is a universal factor.
A closer approximation to ¢(#) is

g(t) = g(t) + O(e'* exp (—2x cosh 2),
— bag? cosh 4 Ua? 5t 42 och box (—
g(t) = \4ar° cosh 4 ++ (4x 67) cosh 4 + 4n° cosh 4 exp (—2z cosh 2),

and again f°. ¢,(#)e"” dé has real roots only. (Pélya gives a second approxi-
mation with {167° cosh 91/4 — 24m cosh 5t/4} (see [8], [12; 45]). This approxi-
mation however is not actually closer than ¢,(£).) We have, namely,

g(t) = 2 (cosh 5t/4){2n° — 34 -+ 4a” cosh t} exp (—27 cosh 2),

so that application of Theorem 22 and of the universal factor cosh 5t/4 gives
the result.

We must not expect that the function Z@) can be approximated by functions
f°. (exp (—2m cosh #))f,(e**' dt where the f, are analytic universal factors.
For if f,() — f( in a circle |£| < e, then f(@ can be continued over the whole
plane. (See [4].) This is not true for the function ¢g(#), which cannot be con-
tinued over the lines Im t = -:}2. The results of the preceding section however
suggest approximations of a different type.

In the following, we shall devote our attention to Ramanujan’s function
instead of the Riemann function. Owing to the pole at s = 1, the formulas
for Riemann’s function show a complication which is probably unessential.

Let z(n) be the coefficients of the power series for the well-known elliptic
modular function,

rs)

(7.6) gy) = yf — yl — xy) = Do r(n)y".


===== tmp/pdfs/o0176-debruijn/page-27.txt =====
222 N. G. DE BRUIJN

With these coefficients Ramanujan [2; Lecture X] constructed his zeta-function
F(s) = do? r(n)/n* = [I {1 — r(p)p™* + py". . The series and the
product are absolutely convergent and ~ 0 for Re s > 13/2. F(s) can be con-
tinued over the whole plane by the formula

re
(7.7) (is) = (20)*°F(s + 6)I(s + 6) = | x **g(e**”) dx.
0
It follows from #°g(e°"*) = x *g(e°*”*) that =Z~(z) satisfies the functional

equation =-(z) = Z (—<). By virtue of the above functional equation of g(z)
we have

age") = {ole}
= et ete) I {1 _ ery (1 _ erly ye

It follows from (7.7) that

(7.8)

(7.9) | e@ = [ ; oe dt,

where

(7.10)  @(#) = (exp (—2z cosh 2) I (1 — ere yy Sg rre yy?

In analogy with the Riemann hypothesis we have the problem as to the reality
of all the roots of (7.9). .Anyhow, the roots lie in the strip | Im z| < 3. We
prefer the discussion of (7.9) to that of (7.3) since ¢ (é) has a simple product
expansion.

There are several aspects connecting the problem concerning the roots of
(7.9) with the results and methods of the preceding sections.

(a). A first indication is given by the domain of regularity and the behaviour
fort > + of ¢ (d), in connection with Theorems 2 and 25. We have namely,
introducing a function g(x),

(7.11) g¢ (t) = (exp (—4n cosh £))q™ (x) (x = e')

where q° (x) is regular for Re x > 0, but cannot be continued over the imaginary
axis. Furthermore g(x) = qg~(1/2*) and q~(a) ~ 2° for Re x > +, so that
¢ (2) is relatively smooth for z-—>© andx-—»0. The nature of 9g” (x) is therefore
related to that of the functions ¢(%) occurring in Theorem 2 (for the special
case P(t) = 4a cosh 1), which lead to trigonometric integrals with a finite number
of non-real roots at most. These g(x) had to be regular in a sector | arg 2| <
4n + 6, 6 > 0, and meromorphic at x = Oand 2 ==.

Té turns out to be important to construct large classes of functions q(x) for
which the finite number of non-real roots is either zero or uniformly bounded.
No such functions have been found yet, apart from polynomials q(x) (see §6).


===== tmp/pdfs/o0176-debruijn/page-28.txt =====
THE ROOTS OF TRIGONOMETRIC INTEGRALS 223

(b). The infinite product in (7.10) reminds of the functions in Theorems 24-27.
Consider

(7.12) Ex@ = / ° (exp (—2x cosh #)) II {1 — Pr" 7d — 7) Pe! dt

We have
el _ erred _ ere’)

)

= (exp (—2zv cosh #))4r°7 [TT {1 + 7k?) + ke}

ma

= 4°y"(exp (—2av cosh 2) I {1 —vk°yY + 47k? cosh’é}.
Hence Zy(z) can be written in the form | |
(7.18) C [ ; Ci) I] d+ fe cosh? de**' dé,

where » runs through an infinite set of real numbers and A > 0. This form
reminds of Theorem 27, although for functions of the type (7.13) we may not ex-
pect a result as general as that theorem. We have to face the rather disappointing
fact that the functions Zy(z) may have an infinity of non-real roots (see (7.18)
below). This does not mean that it is no use considering functions of the type
(7.13), for Z~ (2) can of course also be approximated directly by functions of the
type (7.13) without using Zy(z).

(c). It may be expected that the reality of the roots of a trigonometric in-
tegral is in some way connected with multiplicative properties of the integrand.
Several theorems in §6 point in this direction. Therefore the fact that the inte-.
grand of (7.9) is given in the form of an infinite product may be important.

As an example of a multiplication theorem we give the following one which
is, however, too weak to be of any use here.

THEOREM 28. Let the function ¢,() and ¢.(t) be continuous for —7 <i <a
and O(e**’) for some positive €, (g()* = v:(—2), (¢2(0)* = g2(—d). Suppose
that ~, and ¢, are such that the integrals [*.. e™”’ o;(t)e** dt, 7 = 1, 2, have real
roots only, whatever X, \ > 0, may be. Then the same property holds for the product

1 (t) G2 @.
Proof. For > 0, the product

/ ey, (De at | e*"9,(s)e"** ds


===== tmp/pdfs/o0176-debruijn/page-29.txt =====
224 N. -G. DE BRUIIN

has real roots only. To the right side we apply the universal factor e?”” and
find that also f°. e°” dr f2. e?” oi(A(r — o))g2(4(7 + o)) do has real roots
only. Now making \ ©, we easily deduce that [°. ¢:(i)g.(d)e"” dé has real
roots only. On applying this to e~’’¢,(2) instead of ¢.(f) the same follows
concerning [%.. eg, Oe.(He*** dt.

Functions ¢(é) satisfying the conditions of the theorem are, for mstance,

(7.14) g(t) = exp (—at* — ib — ct? — idt)

where a, b, c, d are real and either a > Oora = 0,b © O0ora=b=0,c>0.
Namely, it follows from Theorem 1 that f°. e“"““"e'*' dt, a > 0, ¢ > 0, has
real roots only; since e”*’, p > 0, is a universal factor we may drop the restriction
about c. The statement concerning (7.14) can now be proved on using a sub-
stitution ¢ = & ++ iq; the cases with a = 0 can be tackled by a limit process.

No items essentially different from (7.14) have been found yet; thus Theorem
28 gave no new trigonometric integrals with real roots only.

(d). Functions of the type (7.12) can be discussed directly by the method de-
veloped in §5. In a simple case we shall find an infinity of non-real roots. The
general case is difficult to deal with but must be expected to show the same
effect.

We consider the function

(7.15) f@= [ ; (exp (—2r cosh od ner |S yet et dt,

and suppose that for a certain integer k > 0 the polynomial P(y) = yo + yy +
+++ + yy” satisfies

(7.16) PQ) =P’) =P") =--- =P?) =0,P (1) #0.
We also introduce the function
(7.17) (8s) = a" fyo tid? + yd + es + ya(Q2n + 1) FH.

We define the coefficients 6, , 8:11, °** by (see (5.3))

n co =]
ert SS tere _ » Be",
y=0 v=k

the terms with e~”’, 0 < » < k, vanish by virtue of (7.16). The following func-
tion plays the same role as H(s) did in §5:

fon)

Hs) = [ e* ps yeu’ du = vs + DI(s + I).


===== tmp/pdfs/o0176-debruijn/page-30.txt =====
THE ROOTS OF TRIGONOMETRIC INTEGRALS 225

The process carried out in §5 now leads to (see (5.7))
(7.18) f@ = SbWGe — NG — 9) + DS pty(—# — yV(-% — »),

from which some information on the roots of f(z) can be obtained.

From (7.18) we can deduce that if ali but a finite number of roots of (7.17)
lie in the half-plane Res > «, then all but a finite number of roots:of f(z) lie
in the strip | Imz| < Max (k + w + 4, 8), forany 6 > 0. Namely, if Imz < 0,
the major contribution to (7.18) is given by Bal(iz — k)T Gz — k).

Now turning to the integrals (7.12) we see that k is large, k = 12N, but it is
not likely that the number w introduced above is < —k&. So probably the
functions (7.12) have an infinity of non-real roots.

In a simple case, namely,

(7.19) fi@ = f (exp (2a cosh 2))(1 — e'\(1 _ ent git dt,

where a > 0, b > 0, we show that an infinity of non-real roots exist. We have
vi(s) = a *° — (a + 0), whose roots are s = 2vri/log (1 + b/a), » = 0, +1,
+2,---. It follows from (7.18) that for |Rez| > 1, Imz < —8,0 < 6 < 3,
we have fi(2) = by,(iz — 1)T (iz — 1) + Of2 "Tz — 1)}. Now, by Rouché’s
theorem, we infer that f,(z) has, from a certain value of | » | onward, just one
root in the neighborhood of any point —7 + 2vm/log (1 + b/a). Apart from
these jf, (2) has but a finite number of roots for Imz < —6.

This failure must not be considered as an argument against the Riemann
hypothesis. Objections of the same “strength” could be used against the true
fact that the roots of [®.. ¢~( cosh 4t e*** dé (see (7.10) and (7.4)) are real.

I express my gratitude to Professor Pélya for his kind interest in the results,
and to Messrs. J. Korevaar and F. van der Blij, of the Mathematisch Centrum,
Amsterdam, for reading the manuscript and indicating many corrections.

REFERENCES

1. N. G. pe Brun, On the zeros of a polynomial and of its derivative, I., Koninklijke Neder-
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(= Indagationes Mathematicae ex Actis Quibus Titulis, Proceedings of the Section
of Sciences, vol. 8(1946), pp. 635-642.

2. G. H. Harpy, Ramanujan; Twelve Lectures on Subjects Suggested by His Life and Work,
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3. J. L. W. V. Jensen, Recherches sur la théorie des équations, Acta Mathematica, vol. 36(1913),
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4. G. Pétya, Algebraische Untersuchungen tiber ganze Funktionen vom Geschlechte- Null und
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5. Guone Péura, Uber die Nullstellen gewisser ganzen Funktionen, Mathematische Zeitschrift,
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6. G. Pénya, Bemerkung tiber die Integraldarstellung der Riemannschen &Funktion, Acta
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===== tmp/pdfs/o0176-debruijn/page-31.txt =====
226 N. G. DE BRUIIN

7. G. Péxya, On the zeros of certain trigonometric integrals, Journal of the London Mathematical
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8. G. Pétya, Uber trigonometrische Integrale mit nur reellen Nullstellen, Journal fir die reine
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9. G. Pérya, Uber die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V.
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MartuematiscH Instituut DER TECHNISCHE HoGEscHOOL
Derr, NETHERLANDS.

