Fourier expansion of the Riemann zeta function and applications

Abstract

TextWe study the distribution of values of the Riemann zeta function ζ(s) on vertical lines ℜs+iR, by using the theory of Hilbert space. We show among other things, that, ζ(s) has a Fourier expansion in the half-plane ℜs≥1/2 and its Fourier coefficients are the binomial transform involving the Stieltjes constants. As an application, we show explicit computation of the Poisson integral associated with the logarithm of ζ(s)−s/(s−1). Moreover, we discuss our results with respect to the Riemann and Lindelöf hypotheses on the growth of the Fourier coefficients. VideoFor a video summary of this paper, please visit https://youtu.be/wI5fIJMeqp4.