Abstract: Ingham (1940) proved that $N(\sigma,T)\ll T^{3(1-\sigma)/(2-\sigma)}\log^{5}{T}$, where $N(\sigma,T)$ counts the number of the non-trivial zeros $\rho$ of the Riemann zeta-function with $\Re\{\rho\}\geq\sigma\geq 1/2$ and $0<\Im\{\rho\}\leq T$. Such estimates are often valuable in the distribution theory of prime numbers. In this talk I will present an explicit version of this result with the exponent $(7-5\sigma)/(2-\sigma)$ of the logarithmic factor. The crucial ingredient in the proof is an explicit estimate with asymptotically correct main term for the fourth power moment of the Riemann zeta-function on the critical line, a result which is of independent interest. This is joint work with Shashi Chourasiya (UNSW Canberra)

Abstract: Moments of L-functions play a fundamental role in number theory. In this talk we will estimate the second moment of L(1/2+it,F x chi), \chi a primitive character mod p^r and F a fixed GL(3) Hecke form averaged over t and chi. As an application we will obtain uniform subconvexity bounds for L(1/2+it, F x chi). This is ongoing joint work with Dasaratharaman, Leung and Pal.