An analogue of Ramanujan's identity for Bernoulli-Carlitz numbers
Su Hu, Min-Soo Kim
TL;DR
The paper extends Ramanujan's identity for odd zeta values to the function-field setting by formulating a Ramanujan-type transformation involving Bernoulli-Carlitz numbers. It builds the analogy through the Carlitz exponential, Bernoulli-Carlitz numbers, and Carlitz factorials, and connects to Drinfeld modular forms via Petrov's A-expansions and Hamahata's xi_n(z) Lambert-series framework. The main result is an explicit identity with αβ equal to the Carlitz period squared, mirroring the classical structure of Ramanujan's formula. This work situates the function-field analogue within a broader modular-analytic context, enriching the arithmetic theory of zeta-like objects over function fields.
Abstract
In his second notebook, Ramanujan discovered the following identity for the special values of $ζ(s)$ at the odd positive integers \begin{equation*}\begin{aligned}α^{-m}\,\left\{\dfrac{1}{2}\,ζ(2m + 1) + \sum_{n = 1}^{\infty}\dfrac{n^{-2m - 1}}{e^{2αn} - 1}\right\} &-(- β)^{-m}\,\left\{\dfrac{1}{2}\,ζ(2m + 1) + \sum_{n = 1}^{\infty}\dfrac{n^{-2m - 1}}{e^{2βn} - 1}\right\}\nonumber &=2^{2m}\sum_{k = 0}^{m + 1}\dfrac{\left(-1\right)^{k-1}B_{2k}\,B_{2m - 2k+2}}{\left(2k\right)!\left(2m -2k+2\right)!}\,α^{m - k + 1}β^k \label{(1.2)},\end{aligned} \end{equation*} where $ α$ and $ β$ are positive numbers such that $ αβ= π^2 $ and $ m $ is a positive integer. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler's famous formula for even zeta values. In this note, we prove an analogue of the above Ramanujan's identity in the functions fields setting, which involves the Bernoulli-Carlitz numbers.