On multiplicative functions with mean-value one on the set of shifted primes

AbstractAn oft-cited result of Peter Shiu bounds the mean value of a nonnegative multiplicative function over a coprime arithmetic progression. We prove a variant where the arithmetic progression is replaced by a sifted set. As an application, we show that the normalized square roots of −1 (mod m) are equidistributed (mod 1) as m runs through the shifted primes q − 1.

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Multiplicative Functions on Shifted Primes

Journal of Number Theory ◽

10.1016/j.jnt.2021.06.027 ◽

2021 ◽

Author(s):

Stelios Sachpazis

Keyword(s):

Multiplicative Functions ◽

Shifted Primes

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Extremal behaviour of ± 1-valued completely multiplicative functions in function fields

Glasnik Matematicki ◽

10.3336/gm.56.1.06 ◽

2021 ◽

Vol 56 (1) ◽

pp. 79-94

Author(s):

Nikola Lelas ◽

Keyword(s):

Rational Function ◽

Finite Fields ◽

Function Fields ◽

Mean Value ◽

Multiplicative Functions ◽

Irreducible Polynomials ◽

Liouville Function ◽

The Mean

We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].

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Integral Limit Laws for Additive Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-1973-017-3 ◽

1973 ◽

Vol 25 (1) ◽

pp. 194-203

Author(s):

J. Galambos

Keyword(s):

Characteristic Function ◽

Fourier Transforms ◽

Mean Value ◽

Characteristic Functions ◽

Asymptotic Formulas ◽

Mean Value Theorem ◽

Multiplicative Functions ◽

Additive Functions ◽

Limit Laws ◽

Integral Limit

In the present paper a general form of integral limit laws for additive functions is obtained. Our limit law contains Kubilius’ results [5] on his class H. In the proof we make use of characteristic functions (Fourier transforms), which reduces our problem to finding asymptotic formulas for sums of multiplicative functions. This requires an extension of previous results in order to enable us to take into consideration the parameter of the characteristic function in question. We call this extension a parametric mean value theorem for multiplicative functions and its proof is analytic on the line of [4].

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