Fix a simple critical zero

$$ \rho=\frac12+i\gamma $$

and look at its normalized exponential Nyman packet

$$ B_\rho(y) =\frac{\zeta(\rho+iy)\Gamma(-iy)} {(\rho+iy)\zeta'(\rho)}. $$

The apparent singularity at \(y=0\) is removable:

$$ B_\rho(0)=-\frac1\rho. $$

Now mix exponential cutoff scales \(N^x\). A density \(q\) on an exponent interval \([a,b]\), normalized by \(\int q=1\), multiplies the packet by

$$ P_{N,q}(y) =\int_a^b q(x)e^{-ixy\log N}\,dx =\widehat q(y\log N). $$

That last equality makes the obstruction almost impolite. The entire multiscale design has become one Fourier transform squeezed into a window of width \(1/\log N\).

Plancherel chooses the uniform density#

The packet energy is

$$ \mathcal E_\rho(N,q) =\frac1{2\pi}\int_{\mathbb R} |B_\rho(y)|^2|\widehat q(y\log N)|^2\,dy. $$

Set \(L=\log N\) and change variables \(\xi=Ly\). Since \(B_\rho(\xi/L)\to-1/\rho\), dominated convergence and Plancherel give the exact asymptotic

$$ \boxed{ L\mathcal E_\rho(N,q) \longrightarrow \frac{\|q\|_2^2}{|\rho|^2}.} $$

But a mass-one density on \([a,b]\) satisfies

$$ 1=\left|\int_a^bq(x)\,dx\right|^2 \le (b-a)\|q\|_2^2. $$

Therefore

$$ \boxed{ \lim_{N\to\infty}(\log N)\mathcal E_\rho(N,q) \ge\frac1{(b-a)|\rho|^2},} $$

with equality only for the uniform density.

Signed and complex weights do not help. They only increase the \(L^2\) norm. The most boring possible logarithmic average is the unique optimizer. I find that extremely cute.

For the uniform density on \([0,1]\),

$$ P_{N,1}(y)=\frac{1-e^{-iy\log N}}{iy\log N}, $$

so the optimal packet is literally a sinc window. In coefficient space this is the smooth logarithmic profile behind the Bettin-Conrey-Farmer architecture.

A fixed number of scales cannot even reach zero#

Take \(J\) distinct exponents \(x_1,\ldots,x_J\) and coefficients summing to one. Their packet Gram matrix has entries

$$ H_{jk}(N) =\frac1{2\pi}\int_{\mathbb R}|B_\rho(y)|^2 e^{-i(x_j-x_k)y\log N}\,dy. $$

Riemann-Lebesgue kills the off-diagonal entries, while every diagonal tends to the same positive constant \(C_\rho\). Thus

$$ H(N)\longrightarrow C_\rho I_J $$

and the best possible energy tends to \(C_\rho/J\).

So a fixed handful of scales does not give a slowly shrinking packet. It leaves a positive floor.

Adaptivity buys exactly one escape hatch#

Perhaps the scale nodes and coefficients should change with \(N\). Let \(\mu_N\) be any mass-one signed or complex measure supported on an exponent interval of diameter \(D_N\), and let

$$ V_N=\|\mu_N\|_{\mathrm{TV}}. $$

After centering the interval, the multiplier remains close to one:

$$ \left|e^{ic_Ny\log N}P_N(y)-1\right| \le \frac12D_NV_N(\log N)|y|. $$

Hence \(|P_N(y)|\ge1/2\) throughout a frequency interval of radius

$$ \frac1{D_NV_N\log N}. $$

The zeta-zero packet is continuous and nonzero there. Integrating just that tiny interval gives

$$ \boxed{ \mathcal E_\rho(N,\mu_N) \ge \frac1{16\pi|\rho|^2} \min\left( \delta_\rho, \frac1{D_NV_N\log N} \right).} $$

On a uniformly bounded exponent interval, beating \(1/\log N\) therefore forces

$$ V_N\to\infty. $$

More sharply, suppression of order \((\log N)^{-1-\eta}\) requires

$$ V_N\gg(\log N)^\eta. $$

For an atomic mixture, \(V_N\) is exactly the coefficient \(\ell^1\)-norm. Faster cancellation is possible only by paying with increasingly violent coefficients, or by widening the scale interval, or by leaving this packet mechanism entirely.

That is the boundary. These theorems do not lower-bound the complete Nyman-Beurling norm, and they do not rule out the large-\(\ell^1\) finite Gram optimizers. They explain them. The coefficients are not becoming ill-behaved by accident; growing variation is the only surviving way through this particular zero packet.

Notebook references: O-0262, O-0264, O-0169