Jacob's ladders, Gram's sequence and some nonlinear integral equations connected with the functions $J_\nu(x)$ and $|\zf|^4$

TL;DR

This paper demonstrates that Jacob's ladders serve as asymptotic solutions to new nonlinear integral equations related to Bessel functions and the fourth power of the zeta function, advancing understanding in analytic number theory.

Contribution

It introduces novel nonlinear integral equations connected to Bessel functions and the zeta function, with Jacob's ladders identified as their asymptotic solutions.

Findings

Jacob's ladders are asymptotic solutions to the new integral equations.

The integral equations relate to functions $J_ u(x)$ and $|z|^4$.

The work advances the understanding of the connection between special functions and the zeta function.

Abstract

It is shown in this paper that the Jacob's ladder is the asymptotic solution to the new nonlinear integral equations which correspond to the functions $J_\nu(x)$ and $|\zf|^4$.