On a function of Ramanujan twisted by a logarithm

TL;DR

This paper derives a two-term modular (functional) relation for a log-twisted Ramanujan Herglotz-type function $\phi_{\log}$ by combining Ramanujan's Lost Notebook modular relation with a derivative-of-Deninger-function framework. It introduces the auxiliary integral $\mathscr{H}(x)$, establishes multiple representations for it, and defines $\mathscr{G}(x)$ to obtain the symmetric relation $\sqrt{\alpha}\mathscr{G}(\alpha)=\sqrt{\beta}\mathscr{G}(\beta)$ for $\alpha\beta=1$, which yields the main two-term equation for $\phi_{\log}$. The proof hinges on contour integration, Mellin transforms, and Ramanujan identities, and relies on an intermediate modular relation for $\mathscr{J}(x)$ and an identity connecting $\phi_1$ and $\phi_0$. The work also links $\mathscr{H}$ to zeta-function data and the auto-correlation function $A(x)$, offering several representations that facilitate connections to Deninger's function and Ramanujan-type transform results, with potential applications to zeta-moment contexts.

Abstract

A two-term functional equation for an infinite series involving the digamma function and a logarithmic factor is derived. A modular relation on page 220 of Ramanujan's Lost Notebook as well as a corresponding recent result for the derivative of Deninger's function are two main ingredients in its derivation. An interesting integral $\mathscr{H}(x)$, which is of independent interest, plays a prominent role in our functional equation. Several alternative representations for $\mathscr{H}(x)$ are obtained.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 2.1
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• proof : Theorem \ref{['log-twist']}
• Lemma 4.1
• proof
• Theorem 4.2