Hybrid moments of the Riemann zeta-function

The "hybrid" moments $$\int_T^{2T}|\zeta(1/2+it)|^k{(\int_{t-G}^{t+G}|\zeta(1/2+ix)|^\ell dx)}^m dt$$ of the Riemann zeta-function $\zeta(s)$ on the critical line $\Re s = 1/2$ are studied. The expected upper bound for the above expression is $O_\epsilon(T^{1+\epsilon}G^m)$. This is shown to be true for certain specific values of the natural numbers $k,\ell,m$, and the explicitly determined range of $G = G(T;k,\ell,m)$. The application to a mean square bound for the Mellin transform function of $|\zeta(1/2+ix)|^4$ is given.Comment: 27 page

Primality testing and Abelian varieties over finite fields

From Gauss to G|del, mathematicians have sought an efficient algorithm to distinguish prime numbers from composite numbers. This book presents a random polynomial time algorithm for the problem. The methods used are from arithmetic algebraic geometry, algebraic number theory and analyticnumber theory. In particular, the theory of two dimensional Abelian varieties over finite fields is developed. The book will be of interest to both researchers and graduate students in number theory and theoretical computer science

finite

On counting and generating curves over smal