On the Sampling of Certain Dirichlet and Prime Sums at Gram Points and their Viscosity
Abstract
Consider the Dirichlet polynomial and its prime-indexed sub-sum ccc E(t) := ∑k≤ [ √( t/2π ) ] (eitlog k)/√k, & ; & E1(t) := ∑substackp≤ [ √(t/2π) ] p prime (eitlog p)/√p. In this work we investigate the statistical and arithmetic properties of the sets 𝒞N := { E(gn):1≤ n≤ N }, 𝒫N := { E1(gn):1≤ n≤ N }, obtained by sampling these sums at the Gram points gn of the Hardy Z-function. We refer to 𝒞N and 𝒫N as the full and prime N-th Gram clouds, respectively. The motivation is that the approximate functional equation expresses both Z(gn) and Z'(gn) through different projections of the same oscillatory phase configuration, suggesting that the geometry of these sampled sums may retain information about the local behavior of nearby zeros. Numerically, we find that Gram-point sampling produces highly non-generic geometric structures: good and isolated bad Gram points occupy markedly different regions of the clouds, the prime cloud of isolated bad points splits into two anisotropic lobes, and the position along the principal direction of the folded prime cloud is strongly correlated with the viscosity of the Hardy Z-function μ(gn)=|Z'(gn)/Z(gn)|, with smallest-viscosity points concentrating near distinguished extremal regions of the lobes.
Related
- Counting Rational Points on a Certain Bihomogeneous Threefold
- Large values of cubic and quartic Dirichlet L-functions of prime moduli
- Extreme values of Dirichlet L-functions on the critical line with a growing prime modulus
- A priori Dirichlet energy bounds for positive Trudinger--Moser critical points on simply connected planar domains
- Gaps, isolated points, and the origin of the ray in the one-dimensional Dirichlet spectrum
BibTeX
@misc{jerby2026sampling,
title = {{On the Sampling of Certain Dirichlet and Prime Sums at Gram Points and their Viscosity}},
author = {Yochay Jerby},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21952},
eprint = {2610.21952},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21952},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Sampling of Certain Dirichlet and Prime Sums at Gram Points and their Viscosity
AU - Yochay Jerby
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21952
UR - https://arxiv.org/abs/2610.21952
AB - Consider the Dirichlet polynomial and its prime-indexed sub-sum ccc E(t) := ∑_k≤ [ √( t/2π ) ] (e^itlog k)/√k, & ; & E_1(t) := ∑_substackp≤ [ √(t/2π) ] p prime (e^itlog p)/√p. In this work we investigate the statistical and arithmetic properties of the sets 𝒞_N := { E(g_n):1≤ n≤ N }, 𝒫_N := { E_1(g_n):1≤ n≤ N }, obtained by sampling these sums at the Gram points g_n of the Hardy Z-function. We refer to 𝒞_N and 𝒫_N as the full and prime N-th Gram clouds, respectively. The motivation is that the approximate functional equation expresses both Z(g_n) and Z'(g_n) through different projections of the same oscillatory phase configuration, suggesting that the geometry of these sampled sums may retain information about the local behavior of nearby zeros. Numerically, we find that Gram-point sampling produces highly non-generic geometric structures: good and isolated bad Gram points occupy markedly different regions of the clouds, the prime cloud of isolated bad points splits into two anisotropic lobes, and the position along the principal direction of the folded prime cloud is strongly correlated with the viscosity of the Hardy Z-function μ(g_n)=|Z'(g_n)/Z(g_n)|, with smallest-viscosity points concentrating near distinguished extremal regions of the lobes.
ER -