Fourier Coefficients of Beurling Functions and a Class of Mellin Transform Formally Determined by its Values on the Even Integers

Abstract

It is a well-known fact that Riemann Hypothesis will follows if the function identically equal to -1 can be arbitrarily approximated in the norm $\norma{.}$ of $L^{2}([0,1],dx)$ by functions of the form $f(x)=\sum_{k=1}^{n}a_{k} ρ(\frac{θ_{k}}{x})$, where $ρ(x)\adef\pfrac{x}$, and $a_{k}\in\cc$, $0<θ_{k}\leqslant 1$ satisfies $\sum_{k=1}^{n}a_{k} θ_{k}=0$. Parsevall Identity $\norma{f(x)+1}^{2}=\sum_{n\in\zz}\modulo{c(n)}^{2}$ is a possible tool to compute or estimate this norm. In this note we give an expression for the Fourier coefficients $c(n)$ of $f+1$, when $f$ is a function defined as above. As an application, we derive an expression for $M_{f}(s)\adef\int_{0}^{1}(f(x)+1) x^{s-1} dx$ as a series that only depends on $M_{f}(2k)$, $k\in\nn$. We remark that the Fourier coefficients $c(n)$ depend on $M_{f}(2k)$ which, for a function $f$ defined as above, can be expressed also in terms of the $a_{k}$&#39;s and $θ_{k}$&#39;s. Therefore, a better control on these parameters will allow to estimate $M_{f}(2k)$ and therefore eventually to handle $\norma{f+1}$ via our expression for the Fourier coefficients and Parsevall Identity.