The connection between the Basel problem and a special integral

TL;DR

This paper establishes a novel connection between the Basel problem, special integrals, and number theory, revealing relationships among zeta function values, Genocchi numbers, Bernoulli numbers, and Bernoulli polynomials.

Contribution

It introduces a new integral representation of the zeta function at 2 and links it to Genocchi and Bernoulli numbers, providing fresh insights into their relationships.

Findings

Zeta function at 2 equals a specific integral.

The special integral is twice another integral.

Derived new relationships between Genocchi and Bernoulli numbers.

Abstract

By using Fubini theorem or Tonelli theorem, we find that the zeta function value at 2 is equal to a special integral. Furthermore, We find that this special integral is two times of another special integral. By using this fact we obtain the relationship between Genocchi numbers and Bernoulli numbers. And get some results about Bernoulli polynomials.