# A new understanding of $\zeta(k)$

## TL;DR

This paper introduces a novel operation in Laurent series to derive explicit series for the Riemann zeta function at positive integers, linking divergent series, Bernoulli numbers, and Borel summation in a new way.

## Contribution

It presents a new method involving Laurent series operations to obtain explicit series for ζ(k), connecting divergent series with Bernoulli numbers and Borel summation.

## Key findings

- Derived explicit series for ζ(k) at positive integers.
- Connected divergent series with Bernoulli numbers via Borel summation.
- Proposed a new approach to understanding ζ-function values.

## Abstract

In this paper, by introducing a new operation in the vector space of Laurent series, the author derived explicit series for the values of $\zeta$-funtion at positive integers, where $\zeta$ denotes the Riemann zeta function. The values of $\zeta(k),\ k>1$ are largely connected with Bernoulli numbers and binomial numbers. The method in this paper seems new, and the resluts are about divergent series. Using Borel summation for these divergent series one can connect $\zeta$ function, Bernoulli numbers, and most series representations of Riemann zeta function.