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

Ali Ebadi

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.

On the Zeros of the Riemann Zeta Function with Two Ordinate 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.
Paper Structure (23 sections, 33 theorems, 379 equations)

This paper contains 23 sections, 33 theorems, 379 equations.

Table of Contents

  1. Introduction and statement of main results
  2. Background
  3. Notations
  4. Main Result
  5. outline of our argument
  6. Preliminaries
  7. The Zeta Function
  8. Estimates with $\mathcal{X}(s)$
  9. Dirichlet L-Functions
  10. Bounds on Various Sums
  11. proof of Theorem 1.1
  12. Preliminaries
  13. Estimation of $I_1$
  14. Bound on $K_1$
  15. Bound on $K_2$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume the Riemann Hypothesis. Let $y \neq 0$ and $C > 0$ be fixed. There exists $T_* = T_*(y,C) > 0$ such that for every $T \geq T_*$, the interval $[T, T(1 + \epsilon)]$ with $\epsilon = T^{-C / \log \log T}$ contains an ordinate $\gamma$ of a zero of $\zeta(s)$ satisfying

Theorems & Definitions (61)

  • Theorem 1.1: Banks2024
  • Theorem 1.2
  • Theorem 2.1
  • Lemma 3.1: Functional Equations
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 51 more