Distribution in coprime residue classes of polynomially-defined multiplicative functions

DOI10.1007/S00209-023-03240-7arXiv2303.14600MaRDI QIDQ6430860

Paul Pollack, Akash Singha Roy

Publication date: 25 March 2023

Abstract: An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F(T)inmathbbZ[T]$ with $f(p)=F(p)$ for all primes $p$. We study the distribution in coprime residue classes of polynomially-defined multiplicative functions, establishing equidistribution results allowing a wide range of uniformity in the modulus $q$. For example, we show that the values $phi(n)$, sampled over integers $nlex$ with $phi(n)$ coprime to $q$, are asymptotically equidistributed among the coprime classes modulo $q$, uniformly for moduli $q$ coprime to $6$ that are bounded by a fixed power of $logx$.

Mathematics Subject Classification ID

Applications of sieve methods (11N36) Arithmetic functions; related numbers; inversion formulas (11A25) Other results on the distribution of values or the characterization of arithmetic functions (11N64)

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