Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2601.15610

On the Zeros of the Riemann Zeta Function with Two Ordinate Shifts

Ali Ebadi
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that for any fixed real numbers y_1, y_2 not equal to 0, and constant C > 0, there exists a threshold T_* = T_*(y_1, y_2, C) > 0 such that for all T >= T_*, the interval [T, T(1 + epsilon)], with epsilon = exp(-C sqrt(log T)), contains at least one gamma satisfying zeta(1/2 + i gamma) = 0, zeta(1/2 + i (gamma + y_1)) != 0, and zeta(1/2 + i (gamma + y_2)) != 0. This extends earlier work by Banks (for a single shift y) to two distinct shifts y_1, y_2. Our argument is based on the behavior of zeta and L functions in zero-free regions via Perron's formula.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ali Ebadi. 2026-01-22. On the Zeros of the Riemann Zeta Function with Two Ordinate Shifts. https://arxiv.org/abs/2601.15610

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

math.NT↗
arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

math.NT↗