Number Theory

    

Asymptotic Distribution of Residuals Within Congruence Classes Generated by Primes

Authors: Gregory M. Sobko

By using the Dirichlet characters for a finite abelian group Gp = Zp=Z/(pZ), where p is a prime, and the corresponding characteristic functions Ф(v(n)), we discuss asymptotic distribution for sums of residuals r = mod(v, p) = [v]p for p from P, a set of all prime numbers. Here vi is a random variable with a certain probability distribution on the set of all natural numbers N, and v(n) is a sum v1 + v2 + … + vn of independent random integers (not necessarily equally distributed). We prove that the residuals of sums [v(n)]p = [v1]p +[v2] + … + [vn]p are asymptotically uniformly distributed on Gp for every prime p ( Gp is a congruence class generated by p ). Then, we prove that the components of the vector of residuals r(v(n))=(r1, r2, …, rpi(n)) are asymptotically independent random variables.Type equation here.

Comments: 10 Pages.

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Submission history

[v1] 2021-09-14 20:13:43

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