# A smaller explicit threshold for decimal digit sums of primes

## Theorem

Let \(s_{10}(n)\) denote the sum of the decimal digits of \(n\). Put

\[
M_{\mathrm{new}}
=138033986655720044641496559943864.
\]

For every integer \(m\ge M_{\mathrm{new}}\) with
\(\gcd(m,9)=1\), there exists a prime

\[
p\le 10^{2m/9}
\]

such that \(s_{10}(p)=m\).

Lehmann's arXiv:2606.04677v1 proves the same statement with

\[
M_{\mathrm{old}}
=177843590339381623423137292296355.
\]

Thus the explicit sufficient threshold decreases by

\[
39809603683661578781640732352491,
\]

or approximately \(22.3846153846154\%\).

## Parameters and exact handoff

We use Lehmann's parameter-uniform theorem with

\[
\eta=\frac{545473}{10000000},
\qquad
\nu=\frac{714321}{2500000},
\qquad
q=10.
\]

Write

\[
c_4=\frac{\log 10}{6},
\qquad
\theta=\frac{c_4}{16(\eta+1/2)}
\]

and choose

\[
c_{43}
=\frac1{200}\frac{c_4}{32}
 \frac{\nu-2\eta}{\eta+1/2}.
\]

The verifier proves exactly that

\[
0<\eta<\nu/2<1/4,\qquad
\nu+2\eta<1/2,\qquad
\nu+\eta<1/2,
\]

and that \(c_{43}\) is strictly inside Lehmann's admissible interval.

Set

\[
Y_\dagger=7.063\times10^{31}
\]

and

\[
L_\dagger
=30674219256826676586999235543080.
\]

Directed Arb arithmetic proves

\[
L_\dagger-1
<
\frac{Y_\dagger}{\log 10}
<
L_\dagger,
\]

so

\[
L_\dagger
=\left\lceil\frac{Y_\dagger}{\log10}\right\rceil.
\]

It also proves

\[
M_{\mathrm{new}}-1
<
\frac92\frac{Y_\dagger}{\log10}+\frac92
<
M_{\mathrm{new}}.
\]

Consequently, for \(m\ge M_{\mathrm{new}}\),

\[
\left\lfloor\frac{2m}{9}\right\rfloor
\ge L_\dagger.
\]

It remains to prove that every source condition is valid for every
integer digit length \(L\ge L_\dagger\). Equivalently, the real
major/minor thresholds must be below \(L\log10\), while the
integer-defined auxiliary thresholds must be at most \(L_\dagger\).
Lehmann packages these conditions in
\[
Y_*=\max\{Y_{41},Y_{42},Y_{43}^*,Y_{45},Y_{47},
Y_{\rm maj},Y_{\rm min},\log x_{\rm AP}\}
\]
but the integer formulation avoids ambiguity from multiplying a
ceiling by \(\log10\).

## Moment-comparison conditions C1--C8

For an integer \(L\ge L_\dagger\), put

\[
u=L^\nu,\quad r=\lceil u\rceil,\quad
L_0=L-2r,\quad
H=10^{\lfloor r/3\rfloor},\quad
K=\lfloor r/2\rfloor.
\]

The endpoint inequality

\[
L>4(u+1)
\]

is permanent because \(L/(u+1)\) is increasing. It implies C1 and
C2. The two exponent comparisons in C3 follow from the six residue
classes of \(r\bmod6\) and

\[
\frac u2>1+\frac{\log9}{\log10}.
\]

C4 follows from \(Y_\dagger>\log4050\), and C6 from
\[
u/3-1>3.
\]

For C8, the permanent sufficient inequality is

\[
1+\log2+\frac{(u+1)\log10}{3}\le u.
\]

Since

\[
H^{-1/2}\le\sqrt{10}\,e^{-c_4u},
\]

C7 follows from

\[
3\cdot10\sqrt{10}<4u.
\]

For C5, split the discrepancy bound into two halves of
\((1/10)e^{-c_4u}\). First,

\[
\frac6H\le60e^{-2c_4u}
\le\frac1{20}e^{-c_4u}
\]

because \(c_4u>\log1200\). Second, C8 and
\[
10^{-K/2}\le\sqrt{10}\,e^{-3c_4u/2}
\]

reduce the other summand to

\[
\frac{24C_{\rm DMR}}{\pi}
u(L\log10)^3\sqrt{10}\,e^{-3c_4u/2}
\le\frac1{20}e^{-c_4u}.
\]

The logarithm of the ratio of the left side to the right side is
strictly negative at \(L_\dagger\). Its derivative with respect to
\(\log L\) is

\[
\nu+3-\frac{c_4\nu}{2}u<0
\]

there and thereafter. Thus C1--C8 hold permanently, so the source's
least integer moment-comparison threshold satisfies
\[
L_{45}\le L_\dagger.
\]

## Characteristic-function conditions C9--C12

Let

\[
\ell=\log L,\qquad
\widetilde D=\frac{2\theta u}{\ell},\qquad
D=2\left\lfloor\frac{\widetilde D}{2}\right\rfloor.
\]

At \(L_\dagger\),

\[
\frac{\theta u}{\ell}>2,\qquad
\widetilde D<L/2,
\]

and the relevant power-to-log ratios are monotone. Hence C9 holds
permanently. C10 follows from the increasing lower bound

\[
\frac{AC}{2\theta}L^{\eta+1/2}\ell>2,
\]

where

\[
A=\frac{400}{\sigma},\qquad
C=2\pi\sigma(\log10)^{\eta-1/2},\qquad
\sigma^2=\frac{33}{4}.
\]

For C11, Lehmann's exact left side is at most \(u\) times

\[
\frac{\nu\ell}{u}
+2\theta\left(
\eta+\frac12+
\frac{\log\ell+\log(eAC/\theta)}{\ell}
\right).
\]

Both variable terms decrease beyond \(L_\dagger\). Directed
evaluation leaves margin

\[
0.0050356070554514\ldots
\]

below \(3c_4/16\).

For C12, the exact left side is at most \(u\) times

\[
2\theta\left(
\frac12+\frac\nu2+
\frac{\tfrac12\log\ell+
      \log A-\tfrac12\log\theta+\tfrac12}{\ell}
\right).
\]

This coefficient is also decreasing, and its endpoint margin below
\(c_4\) is

\[
0.3172384267495728\ldots.
\]

Thus C9--C12 are permanent from \(L_\dagger\).

## Signed floor envelope for C13

The first C13 inequality is immediate and permanent:

\[
\left(\frac{3c_4}{4}-c_{43}\right)u-\log7>0.
\]

The second inequality is the only place where we sharpen the source's
closed-form certificate. Put

\[
T=\eta\ell+\log C,\qquad
W=\widetilde D-2
\]

and define

\[
h(z)=(z-1)T+\frac z2-\frac z2\log z.
\]

After moving the right side to the left, exact C13b is

\[
\log7+c_{43}u+h(D)\le0.
\tag{1}
\]

Because

\[
W<D\le\widetilde D
\]

and

\[
\frac{\partial h}{\partial z}
=T-\frac12\log z,
\]

it is enough to show this derivative is negative throughout the
floor interval. At \(L_\dagger\),

\[
T-\frac12\log W<0.
\]

Moreover, differentiating with respect to \(\ell\) gives

\[
\frac{d}{d\ell}
\left(T-\frac12\log W\right)
<
\eta-\frac12\left(\nu-\frac1\ell\right)<0.
\]

Therefore \(h(D)\le h(W)\) for every \(L\ge L_\dagger\). Define the
continuous upper envelope

\[
E(L)=\log7+c_{43}u+h(W).
\]

Directed evaluation gives

\[
E(L_\dagger)
<-14.5916228525416.
\tag{2}
\]

It remains to prove that \(E\) decreases. Let

\[
d=\frac\nu2-\eta,\qquad
a=2\theta d-c_{43},\qquad
K_c=\log C-\frac12\log(2\theta).
\]

The elementary inequality

\[
-\frac12\log\left(1-\frac2{\widetilde D}\right)
\le\frac1W
\]

gives

\[
T-\frac12\log W
\le
-d\ell+\frac12\log\ell+K_c+\frac1W.
\]

Using
\[
W'=\widetilde D\left(\nu-\frac1\ell\right)
\]

in the exact derivative of \(E\), and retaining the negative leading
term, yields

\[
\frac1u\frac{dE}{d\ell}
\le
-\nu a+\frac{\theta\nu}{\ell}
+\frac{2\theta\nu}{\ell}
\left(\frac12\log\ell+K_c+\frac1W\right).
\tag{3}
\]

The positive remainder on the right of (3) decreases: \(W\)
increases, and the bracket is greater than \(1/2\), so its quotient
by \(\ell\) has negative derivative. At \(L_\dagger\), the reserve in
(3) is

\[
2.9061913241106\ldots\times10^{-9}>0.
\]

Hence \(E'(L)<0\) for every \(L\ge L_\dagger\). Together with (2),
this proves exact C13b permanently, without the source paper's
absolute-value floor remainder.

Therefore the source's least integer characteristic-function
threshold satisfies
\[
L_{43}^*\le L_\dagger.
\]

## Remaining thresholds

The major-arc criterion is

\[
C^{(1)}Y^{-1/2+\nu+2\eta}
+2C^{(0)}Y^{\eta-1}
+8\pi Y^{-1/2+2\eta}
<
\frac{(c_0-\tfrac1{40})\sqrt{\log10}}
     {\sqrt{2\pi\sigma^2}}.
\]

All three exponents are negative. At \(Y=Y_\dagger\), directed Arb
evaluation gives margin

\[
2.84673134769017\ldots\times10^{-6}.
\]

Thus the inequality remains true for every larger \(Y\), proving
\(Y_{\rm maj}<Y_\dagger\).

The minor-arc criterion is checked in logarithmic form:

\[
\log(C_{\rm min}/9)+\frac92\log Y
-81c_1Y^{2\eta}
<
\log\frac{\sqrt{\log10}}{240\sqrt{2\pi\sigma^2}}.
\]

At \(Y_\dagger\), the margin is

\[
0.0044753021567077\ldots.
\]

Its left side is decreasing because

\[
162\eta c_1Y_\dagger^{2\eta}-\frac92>0,
\]

and this reserve grows with \(Y\). Hence
\(Y_{\rm min}<Y_\dagger\).

The same directed certificate proves

\[
Y_{41},Y_{42},\log x_{\rm AP}<Y_\dagger.
\]

For \(Y_{47}\), it checks the source's unrounded digit-length power
plus one below \(L_\dagger\), which rigorously proves that its integer
ceiling is strictly below \(L_\dagger\). The C1--C8 proof above gives
\(L_{45}\le L_\dagger\), and the C9--C13 proof gives
\(L_{43}^*\le L_\dagger\).

Thus every real threshold is below \(Y_\dagger\), and every
integer-defined auxiliary digit length is at most \(L_\dagger\).

## Conclusion

Let \(m\ge M_{\rm new}\) be coprime to \(9\), and set

\[
L=\left\lfloor\frac{2m}{9}\right\rfloor,\qquad x=10^L.
\]

The exact handoff gives \(L\ge L_\dagger\), hence
\(\log x=L\log10>Y_\dagger\). The preceding integer and real
threshold checks show directly that every hypothesis in Lehmann's
parameter-uniform major/minor-arc positivity theorem holds. It follows
that there is a prime

\[
p\le x\le10^{2m/9}
\]

with \(s_{10}(p)=m\), as claimed.

## Independent reconstruction

The primary checker uses Python-FLINT/Arb and is rerun at 256, 384,
and 512 bits. A separate Julia/Nemo checker retypes the C11, C12,
C13, \(Y_{47}\), major-arc, minor-arc, and integer-ceiling
calculations. In particular it reconstructs the major-arc right side
from Lehmann's certified bound

\[
c_0>1-1.3\times10^{-32}
\]

rather than evaluating the error function. Both directed
implementations obtain positive reserves.
