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

TL;DR

This paper proves the existence of zeros of the Riemann zeta function within specific intervals that are shifted by two fixed real numbers, extending previous results from one to two shifts using Perron's formula.

Contribution

It extends earlier work by Banks to show zeros with two distinct ordinate shifts, providing new insights into the distribution of zeta zeros in shifted regions.

Findings

Zeros exist in small intervals around large T with two fixed shifts

Interval length is exponentially small in sqrt(log T)

Generalizes single-shift results to two shifts

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.