Functional equation for Mellin transform of Fourier series associated with 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 to certain Dirichlet series and modular forms satisfy the same functional equation.