On certain Fourier expansions for the Riemann zeta function

TL;DR

This work develops Fourier expansions for the Riemann zeta function and related $L$-functions within the $L^2(\mu)$ framework, deriving explicit coefficient formulae tied to zeta zeros and derivatives. A key contribution is the contour-residue derivation of coefficients for $1/\zeta(s)$ and its connections to the locations of nontrivial zeros, along with a new Mellin-transform based expression for the Fourier coefficients of the Riemann $\xi$-function, including a Whittaker-function representation for the coefficients. The paper also analyzes partial sums via Fejér kernels, showing conditional convergence results under the Riemann Hypothesis and discussing the practical constraints posed by potential discontinuities. The concluding discussion links absolute convergence of these Fourier series to Wiener's theorem and to RH, outlining implications for the analytic structure of $1/\zeta(\sigma+it)$ in the critical strip and suggesting further avenues to exploit Mellin and residue techniques in this zeta-analytic setting.

Abstract

We build on a recent paper on Fourier expansions for the Riemann zeta function. We establish Fourier expansions for certain $L$-functions, and offer series representations involving the Whittaker function $W_{γ,μ}(z)$ for the coefficients. Fourier expansions for the reciprocal of the Riemann zeta function are also stated. A new expansion for the Riemann xi function is presented in the third section by constructing an integral formula using Mellin transforms for its Fourier coefficients.

Theorems & Definitions (14)

• Theorem 1.1
• Corollary 1.1.1
• Theorem 1.2
• Lemma 2.1
• proof : Proof of Theorem \ref{['thm:thm1']}
• proof : Proof of Corollary \ref{['corollary:Cor1']}
• proof : Proof of Theorem \ref{['thm:thm2']}
• Theorem 3.1
• proof
• Corollary 3.1.1