Fake mu’s

Abstract

Let ϝ ( n ) \digamma (n) denote a multiplicative function with range { − 1 , 0 , 1 } \{-1,0,1\} , and let F ( x ) = ∑ n = 1 ⌊ x ⌋ ϝ ( n ) F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \digamma (n) . Then F ( x ) / x = a x + b + E ( x ) F(x)/\sqrt {x} = a\sqrt {x} + b + E(x) , where a a and b b are constants and E ( x ) E(x) is an error term that either tends to 0 0 in the limit or is expected to oscillate about 0 0 in a roughly balanced manner. We say F ( x ) F(x) has persistent bias b b (at the scale of x \sqrt {x} ) in the first case, and apparent bias b b in the latter. For example, if ϝ ( n ) = μ ( n ) \digamma (n)=\mu (n) , the Möbius function, then F ( x ) = ∑ n = 1 ⌊ x ⌋ μ ( n ) F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \mu (n) has apparent bias 0 0 , while if ϝ ( n ) = λ ( n ) \digamma (n)=\lambda (n) , the Liouville function, then F ( x ) = ∑ n = 1 ⌊ x ⌋ λ ( n ) F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \lambda (n) has apparent bias 1 / ζ ( 1 / 2 ) 1/\zeta (1/2) . We study the bias when ϝ ( p k ) \digamma (p^k) is independent of the prime p p , and call such functions fake μ ′ s \mu ’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. For such a function F ( x ) F(x) with apparent bias b b , we also show that F ( x ) / x − a x − b F(x)/\sqrt {x}-a\sqrt {x}-b changes sign infinitely often.