Some conjectures on the zeros of approximates to the Riemann $\Xi$-function and incomplete gamma functions

TL;DR

This paper explores the zeros of truncated sums of the Riemann $\Xi$-function involving incomplete gamma functions, proposing conjectures that could imply the Riemann Hypothesis and supporting them with computational evidence.

Contribution

It introduces a new conjecture about the zeros of truncated sums of the $\Xi$-function and discusses how this could imply the Riemann Hypothesis, supported by computational data.

Findings

Conjecture that zeros of $\Xi_N(z)$ have nondecreasing imaginary parts in the first quadrant.

The conjecture implies the Riemann Hypothesis if true.

Computational evidence supports the conjecture and related hypotheses.

Abstract

Riemann conjectured that all the zeros of the Riemann $\Xi$-function are real, which is now known as the Riemann Hypothesis (RH). In this article we introduce the study of the zeros of the truncated sums $\Xi_N(z)$ in Riemann's uniformly convergent infinite series expansion of $\Xi (z)$ involving incomplete gamma functions. We conjecture that when the zeros of $\Xi_N (z)$ in the first quadrant of the complex plane are listed by increasing real part, their imaginary parts are monotone nondecreasing. We show how this conjecture implies the RH, and discuss some computational evidence for this and other related conjectures.