Fourier Expansion of the Riemann zeta function and applications

We study the distribution of values of the Riemann zeta function $\zeta (s)$ on vertical lines $\mathrm{\Re }s+i\mathbb{R}$, by using the theory of Hilbert space. We show among other things, that, $\zeta (s)$ has a Fourier expansion in the half-plane $\mathrm{\Re }s\ge 1\mathrm{/}2$ 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 $\zeta (s)-s\mathrm{/}(s-1)$. Moreover, we discuss our results with respect to the Riemann and Lindel"{o}f hypotheses on the growth of the Fourier coefficients.