A. Sankaranarayanan
TIFR, Mumbai
May 10, 2018
Riemann zeta-function and some conjectures: Assuming the simplicity of the zeros of the Riemann zeta function $\zeta(s)$, Gonek and Hejhal studied the sum \[ J_{-k}(T) := \sum_{0 < \gamma \le T} |\zeta'(\rho)|^{-2k} \] for real number $k \ge 0$ and conjectured that $J_{-k}(T)\asymp T (\log T)^{(k-1)^2}$ for any $k in \mathbb R$. Assuming Riemann hypothesis and $J_{-1}(T) \ll T$, Ng (NN, PLMS (2004)) proved that the Mertens function $M(x) \ll \sqrt{x}(log x)^{3/2}$. He also pointed out that with the additional hypothesis of $J_{-\frac{1}{2}}(T) \ll T(\log t)^{1/4}$ one gets $M(x) \ll \sqrt{x}(\log x)^{5/4}$. Here we show that it is possible to obtain $M(x) \ll \sqrt{x}(\log x)^{a}$ for any real number $a \in [5/4,3/2]$, under similar hypotheses. This is a joint work with Dr. Biswajyoti Saha.
TIFR, Mumbai
May 10, 2018
Riemann zeta-function and some conjectures: Assuming the simplicity of the zeros of the Riemann zeta function $\zeta(s)$, Gonek and Hejhal studied the sum \[ J_{-k}(T) := \sum_{0 < \gamma \le T} |\zeta'(\rho)|^{-2k} \] for real number $k \ge 0$ and conjectured that $J_{-k}(T)\asymp T (\log T)^{(k-1)^2}$ for any $k in \mathbb R$. Assuming Riemann hypothesis and $J_{-1}(T) \ll T$, Ng (NN, PLMS (2004)) proved that the Mertens function $M(x) \ll \sqrt{x}(log x)^{3/2}$. He also pointed out that with the additional hypothesis of $J_{-\frac{1}{2}}(T) \ll T(\log t)^{1/4}$ one gets $M(x) \ll \sqrt{x}(\log x)^{5/4}$. Here we show that it is possible to obtain $M(x) \ll \sqrt{x}(\log x)^{a}$ for any real number $a \in [5/4,3/2]$, under similar hypotheses. This is a joint work with Dr. Biswajyoti Saha.