Ramanujan Summation and the Exponential Generating Function $ \sum_{k=0}^{\infty}\frac{z^{k}}{k!}\zeta ^{\prime}(-k)$

TL;DR

This paper explores Ramanujan summation of specific divergent series related to exponential generating functions and establishes connections to the derivatives of the Riemann zeta function at negative integers, linking Ramanujan and Borel summation methods.

Contribution

It calculates Ramanujan sums for particular exponential generating functions and reveals a novel relation involving the derivatives of the Riemann zeta function and Euler sums.

Findings

Derived explicit Ramanujan sums for exponential generating functions.

Discovered a surprising relation connecting zeta function derivatives to Euler sums.

Expressed Ramanujan summation results in terms of Borel summation.

Abstract

In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from which follows a formula that connects the derivatives of the Riemann zeta - function at the negative integers to the Ramanujan summation of the divergent Euler sums $\sum_{n\ge 1} n^kH_n, k \ge 0$, where $H_n= H_n^{(1)}$. Further, we express our results on the Ramanujan summation in terms of the classical summation process called the Borel sum.