On all real zeros for a new class of the even entire function

TL;DR

This paper introduces a new class of even entire functions related to products and series with real coefficients, providing conditions for all zeros to be real and addressing longstanding mathematical problems.

Contribution

It establishes a sufficient condition for all zeros of this new class of functions to be real and offers solutions to open problems posed by Lagarias and Montague.

Findings

Provided a sufficient condition for all zeros to be real

Solved a problem posed by Lagarias and Montague

Suggested open problems for further research

Abstract

In this article we propose a new class of the even entire function connected with the product and series with the real coefficients. We address a sufficient condition for all real zeros for it. As a typical example, we give an answer to the problem of Lagarias and Montague. We suggest the open problems for the class of the even entire function.