Large zeta sums and zeros of the Riemann zeta function
Abstract
For real t and x≥ 1, set S(x,t)=∑n≤ x nii t. We prove an unconditional inverse theorem relating large values of S(x,t), with |t| large, to zeros of the Riemann zeta function near height t. More precisely, if T≤ |t|≤ 2T, exp(√(log T))≤ x≤ √T, and |S(x,t)|=x/N with N≤ (log x)1/100, then for every cN6≤ L≤ (log x)/2 a disk centered at 1+iiϕ, where |ϕ−t|≪ N, contains at least L/360 zeros of ζ(s). As a consequence, a quantitative restriction on zeros in a short family of arbitrarily thin fixed windows to the left of the line Ree s=1 yields S(x,t)≪ x/(log x)1/100 in polynomial ranges of x. The proof adapts the zero-forcing mechanism of Granville and Soundararajan for large character sums. In the zeta setting the spectral height is shifted by t, and an additional residue from the pole of ζ(s) appears in the Gaussian transform; in the range considered here that residue is exponentially small.
Related
- Oscillation of partial sums of the Möbius function and zeros of Riemann's zeta function
- More than 8/9 zeros of the Riemann zeta function are simple or on the critical line
- One-level density of zeros of the Riemann zeta function weighted by its special values
- Half-sized gaps between zeros of the Riemann zeta function
- Simple critical zeros and distinct zeros of the Riemann zeta function in short intervals
BibTeX
@misc{dong2026large,
title = {{Large zeta sums and zeros of the Riemann zeta function}},
author = {Zikang Dong and Ruihua Wang and Weijia Wang and Hao Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.31060},
eprint = {2608.31060},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.31060},
}
RIS
TY - UNPB
TI - Large zeta sums and zeros of the Riemann zeta function
AU - Zikang Dong
AU - Ruihua Wang
AU - Weijia Wang
AU - Hao Zhang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.31060
UR - https://arxiv.org/abs/2608.31060
AB - For real t and x≥ 1, set S(x,t)=∑_n≤ x n^ii t. We prove an unconditional inverse theorem relating large values of S(x,t), with |t| large, to zeros of the Riemann zeta function near height t. More precisely, if T≤ |t|≤ 2T, exp(√(log T))≤ x≤ √T, and |S(x,t)|=x/N with N≤ (log x)^1/100, then for every cN^6≤ L≤ (log x)/2 a disk centered at 1+iiϕ, where |ϕ−t|≪ N, contains at least L/360 zeros of ζ(s). As a consequence, a quantitative restriction on zeros in a short family of arbitrarily thin fixed windows to the left of the line Ree s=1 yields S(x,t)≪ x/(log x)^1/100 in polynomial ranges of x. The proof adapts the zero-forcing mechanism of Granville and Soundararajan for large character sums. In the zeta setting the spectral height is shifted by t, and an additional residue from the pole of ζ(s) appears in the Gaussian transform; in the range considered here that residue is exponentially small.
ER -