Fake Mu's

Abstract

Let ϝ(n) denote a multiplicative function with range {−1, 0, 1}, and let, 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 ϝ(n) = µ(n), the Möbius function, thenWe study the bias when ϝ(p k ) 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.

On the average value of $π(t)-\text{li}(t)$

“…(c) The bias of λ(n) and related functions [Hum13], [MMT21]. One could also attempt to extend our results to more general number fields.…”

On the average value of

“…(b) The Chebyshev bias for primes in arithmetic progressions [23], [10]. (c) The bias of 𝜆(𝑛) and related functions [11], [17].…”