2021/12/09 by Greg S. Martin, Martin, Greg, Michael J. Mossinghoff +3
Mathematics · #Analytic Number Theory Research #Benford’s Law and Fraud Detection #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2112.05227
Let f(n) denote a multiplicative function with range \-1,0,1\, and let F(x) = ∑n≤ x f(n). Then F(x)/√(x) = a√(x) + b + E(x), where a and b are constants and E(x) is an error term that either tends to 0 in the limit, or is expected to oscillate about 0 in a roughly balanced manner. We say F(x) has persistent bias b (at the scale of √(x)) in the first case, and apparent bias b in the latter. For example, if f(n)=μ(n), the Möbius function, then F(x) = ∑n≤ x μ(n) has b=0 so exhibits no persistent or apparent bias, while if f(n)=λ(n), the Liouville function, then F(x) = ∑n≤ x λ(n) has apparent bias b=1/ζ(1/2). We study the bias when f(pk) is independent of the prime p, and call such functions fake μ's. We investigate the conditions required for such a function to exhibit a persistent or apparent bias, determine the functions in this family with maximal and minimal bias of each type, and characterize the functions with no bias of either type. For such a function F(x) with apparent bias b, we also show that F(x)/√(x)-a√(x)-b changes sign infinitely often.