On The Zeros of a Class of Dirichlet series

Let $Q(x,y)=ax^{2}+bxy+cy^{2}$ be a real and positive definite quadratic form. The classical Epstein zeta function is defined as the Dirichlet series
Z_{2}(s,\,Q)=\sum_{m,n\neq0}\frac{1}{Q(m,n)^{s}},\,\,\,\,\,\text{Re}(s)>1,\label{Classical Epstein}
where the notation given in the subscript $m,\,n\neq0$ means that only the term $m=n=0$ is omitted from the infinite series.

In 1949, S. Chowla and A. Selberg announced the following formula for (\ref{Classical Epstein}), valid in the entire complex plane,
a^{s}\Gamma(s)\,Z_{2}(s,\,Q) & =2\Gamma(s)\zeta(2s)+2k^{1-2s}\pi^{1/2}\Gamma\left(s-\frac{1}{2}\right)\zeta(2s-1)\nonumber \\
 & \,\,\,\,\,\,+\,8k^{1/2-s}\pi^{s}\sum_{n=1}^{\infty}n^{s-1/2}\sigma_{1-2s}(n)\,\cos\left(n\pi b/a\right)K_{s-1/2}\left(2\pi k\,n\right),\label{Selberg Chowla Formula}
where $d:=b^{2}-4ac$ is the discriminant of the quadratic form, $k^{2}:=|d|/4a^{2}$ and $\sigma_{\nu}(n)=\sum_{d|n}d^{\nu}$ is the generalized divisor function of index $\nu$. Also, $\zeta(s)$ denotes the classical Riemann zeta function and $K_{\nu}$ the modified Bessel function.

By using no more than basic tools of Complex and Fourier analysis, in this talk we will discuss generalizations of (\ref{Selberg Chowla Formula}) for large classes of Dirichlet series. In particular, we will see how the representation (\ref{Selberg Chowla Formula}) is connected with the infinitude of zeros of $\zeta(s)$ at the critical line $\text{Re}(s)=\frac{1}{2}$ (Hardy's Theorem). As a very simple application of our method, we will argue that the conclusion of Hardy's Theorem for $\zeta(s)$ can be deduced from Jacobi's 4-square Theorem, which constitutes a curious connection between a purely analytic theorem involving deep properties of $\zeta(s)$ and an interesting arithmetical property concerning the representation of a given integer as the sum of four squares.


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Pedro Ribeiro

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CMUP Informal PHD Seminar