Biases in prime factorizations and Liouville functions for arithmetic progressions

Humphries, Peter; Shekatkar, Snehal M.; Wong, Tian An

doi:10.5802/jtnb.1066

Cited by 2 publications

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“…We remark that biases and asymptotic behavior of other generalizations of µ(n) and λ(n) occur in the literature. For example, Humphries, Shekatkar, and Woo [6] recently studied the summatory function of λ(n; q, a) = (−1) Ω(n;q,a) , where Ω(n; q, a) denotes the number of prime factors p of n (counting multiplicity) which satisfy p ≡ a mod q. Also, biases exhibited in families of weighted sums, such as L α (x) =…”

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confidence: 99%

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Fake Mu's

Martin¹,

Mossinghoff²,

Trudgian³

2021

Preprint

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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.

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confidence: 99%

“…We can also construct an example of a function from part (iii) of Theorem 1 with no apparent bias, by selecting α = 1/ √ 2 and β = 1 in (6). That is, we select the values δ 2j+1 for j ≥ 1 according to the binary expansion of 1/ √ 2 = (0.1011010100 .…”

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confidence: 99%

Fake Mu's

Martin¹,

Mossinghoff²,

Trudgian³

2021

Preprint

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Fake mu’s

Martin

Mossinghoff

Trudgian

2023

Proc. Amer. Math. Soc.

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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.

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Biases in prime factorizations and Liouville functions for arithmetic progressions

Cited by 2 publications

References 20 publications

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