Fake mu's

with Michael J. Mossinghoff and Timothy S. Trudgian

Let f(n) denote a multiplicative function with range {-1,0,1}, and let F(x) = Σnx f(n). Then F(x)/√x = ax + 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) = Σnx μ(n) has b = 0 so exhibits no persistent or apparent bias, while if f(n) = λ(n), the Liouville function, then F(x) = Σnx λ(n) has apparent bias b = 1/ζ(½). 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)/√xaxb changes sign infinitely often.

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