Mean-Value of Product of Shifted Multiplicative Functions and Average Number of Points on Elliptic Curves

R. Balasubramanian, Sumit Giri

Published 2014-08-14, updated 2014-09-03Version 3

In this paper, we consider the mean value of the product of two real valued multiplicative functions with shifted arguments. The functions $F$ and $G$ under consideration are close to two nicely behaved functions $A$ and $B$, such that the average value of $A(n-h)B(n)$ over any arithmetic progression is only dependent on the common difference of the progression. We use this method on the problem of finding mean value of $K(N)$, where $K(N)/\log N$ is the expected number of primes such that a random elliptic curve over rationals has $N$ points when reduced over those primes.