Colloquium abstracts

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'( ho)|^{-2k} ] for real number $k ge 0$ and conjectured that $J_{-k}(T)asymp T (log T)^{(k-1)^2}$ for any $k in R$. 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.