Functional equation for Mellin transform of Fourier series associated with modular forms

TL;DR

This paper investigates the Mellin transforms of functions related to Fourier series, extending Ramanujan's classical functional equation, and demonstrates their connection to Dirichlet L-functions and modular forms.

Contribution

It generalizes Ramanujan's functional equation for Mellin transforms, linking it to Fourier series of modular forms and Dirichlet L-functions.

Findings

Mellin transforms satisfy a generalized functional equation involving a parameter k.

The Mellin transforms of Fourier series associated with modular forms meet the new functional equation.

Connections established between Mellin transforms, Dirichlet L-functions, and modular forms.

Abstract

Let $X_1(s)$ and $X_2(s)$ denote the Mellin transforms of $\chi_{1}(x)$ and $\chi_{2}(x)$, respectively. Ramanujan investigated the functions $\chi_1(x)$ and $\chi_2(x)$ that satisfy the functional equation \begin{equation*} X_{1}(s)X_2(1-s) = \lambda^2, \end{equation*} where $\lambda$ is a constant independent of $s$. Ramanujan concluded that elementary functions such as sine, cosine, and exponential functions, along with their reasonable combinations, are suitable candidates that satisfy this functional equation. Building upon this work, we explore the functions $\chi_1(x)$ and $\chi_2(x)$ whose Mellin transforms satisfy the more general functional equation \begin{equation*} \frac{X_1(s)}{X_2(k-s)} = \sigma^2, \end{equation*} where $k$ is an integer and $\sigma$ is a constant independent of $s$. As a consequence, we show that the Mellin transform of the Fourier series associated with certain Dirichlet L-functions and modular forms satisfy the same functional equation.