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

TL;DR

This paper studies the mean value of products of shifted multiplicative functions and applies it to estimate the average number of primes for which a random elliptic curve has a specified number of points, combining number theory and elliptic curve analysis.

Contribution

It introduces a method to evaluate mean values of products of shifted multiplicative functions and applies it to elliptic curve point count distribution.

Findings

Derived formulas for mean values of shifted multiplicative functions.

Estimated the average number of primes with a given elliptic curve point count.

Connected multiplicative function analysis with elliptic curve prime distributions.

Abstract

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.