On the Mellin transforms of powers of Hardy's function.

Aleksandar Ivić (1)

1 Katedra Matematike

Various properties of the Mellin transform function $$\mathcal{M}_k(s):= \int_1^{\infty} Z^k(x)x^{-s}\,dx$$ are investigated, where $$Z(t):=\zeta(\frac{1}{2}+it)\,\chi(\frac{1}{2}+it)^{-1/2},~~~~\zeta(s)=\chi(s)\zeta(1-s)$$ is Hardy's function. Connections with power moments of $|\zeta(\frac{1}{2}+it)|$ are established, and natural boundaries of $\mathcal{M}_k(s)$ are discussed.

Mots clés: Hardy's function; Mellin transforms; Riemann zeta-function; analytic continuation; absolute convergence

Domaines: Mathématiques [math]

Aleksandar Ivić. On the Mellin transforms of powers of Hardy's function.. Hardy-Ramanujan Journal, 2010, Volume 33 - 2010, pp.32 - 58. ⟨10.46298/hrj.2010.170⟩. ⟨hal-01112545⟩