# Representations of the Riemann zeta function: A probabilistic approach

## TL;DR

This paper presents a probabilistic method to derive integral representations and Euler's recurrence for the Riemann zeta function at positive integers, offering an elementary proof using logistic distribution properties.

## Contribution

It introduces a novel probabilistic approach to analyze the Riemann zeta function, providing simpler proofs and new integral representations.

## Key findings

- Elementary proof of Euler's recurrence formula
- Integral representations at positive and fractional points
- Use of logistic distribution in zeta function analysis

## Abstract

In this paper, we give a short elementary proof of the well known Euler's recurrence formula for the Riemann zeta function at positive even integers and integral representations of the Riemann zeta function at positive integers and at fractional points by means of probabilistic approach. The proof is based on the moment generating function and the characteristic function of logistic and half-logistic distributions in probability theory.