On the residue class distribution of the number of prime divisors of an integer

2009/06/05 by Michael James Coons, Michael Coons, Coons, Michael +2
Computer Science · Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Coding theory and cryptography #math.NT #msc:11N37 #msc:11N60

paper · pdf · doi:10.48550/arxiv.0906.1029

7 pages

arxiv created 2009/06/05 · arxiv updated 2009/12/01

Abstract

The \em Liouville function is defined by \gl(n):=(-1)^Ω(n) where Ω(n) is the number of prime divisors of n counting multiplicity. Let \z_m:=e^2πi/m be a primitive m--th root of unity. As a generalization of Liouville's function, we study the functions \gl_m,k(n):=\z_m^kΩ(n). Using properties of these functions, we give a weak equidistribution result for Ω(n) among residue classes. More formally, we show that for any positive integer m, there exists an A>0 such that for all j=0,1,...,m-1, we have #\n≤ x:Ω(n)≡ j (\bmod m)\=(x)/(m)+O((x)/(log^A x)). Best possible error terms are also discussed. In particular, we show that for m>2 the error term is not o(x^\ga) for any \ga<1.

Related

• Residue races of the number of prime divisors function
• (Non)Automaticity of number theoretic functions
• Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions
• Application of an accurate remainder term in the calculation of Residue Class Distributions
• Mean values of multiplicative functions and applications to the distribution of the sum of divisors