Zeros of combinations of the Riemann $\Xi$-function and the confluent hypergeometric function on bounded vertical shifts

TL;DR

This paper demonstrates that a series of bounded vertical shifts of a product involving the Riemann $\Xi$-function and a confluent hypergeometric function has infinitely many zeros on the critical line, extending Hardy's classical result.

Contribution

It introduces a new integral representation linking the Riemann $\Xi$-function and hypergeometric functions, proving the existence of infinitely many zeros on the critical line for a broader class of functions.

Findings

Series of shifted products has infinitely many zeros on the critical line.

Generalizes Hardy's theorem to new functions involving hypergeometric components.

Uses integral representations and theta transformation techniques.

Abstract

In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral representation involving the Riemann $\Xi$-function and the confluent hypergeometric function linked to the general theta transformation. Using this result, we show that a series consisting of bounded vertical shifts of a product of the Riemann $\Xi$-function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line, thereby generalizing a previous result due to the first and the last authors along with Roy and Robles. The latter itself is a generalization of Hardy's theorem.