Oberseminar ZAHLENTHEORIE

Athanasios Sourmelidis (TU Graz, Austria)

Continuous and Discrete Means of Hardy's $Z$-function

Abstract. On the pursuit of proving the Riemann Hypotheses, Hardy and Littlewood were the first who proved that infinitely many of the complex zeros of the Riemann zeta-function $\zeta$ lie on the vertical line $1/2+i\mathbb{R}$. Their method was based on studying the integral of a variant of $\zeta$, the so-called Hardy's $Z$-function. A few years later Titchmarsh proposed a different way on proving the same result by studying sums of the $Z$-function over specific points of the line $1/2+i\mathbb{R}$. After introducing the required background and machinery, we will see that these two methods are, in fact, equivalent. Moreover, it will be shown that this equivalence holds for a wider class of zeta- and $L$-functions satisfying certain axioms.