Distribution in coprime residue classes of polynomially-defined multiplicative functions
Abstract
An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F(T)\in \mathbb{Z}[T]$ 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 $n \le x$ 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 $\log{x}$.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2023
- DOI:
- 10.48550/arXiv.2303.14600
- arXiv:
- arXiv:2303.14600
- Bibcode:
- 2023arXiv230314600P
- Keywords:
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- Mathematics - Number Theory;
- Primary 11A25;
- Secondary 11N36;
- 11N64
- E-Print:
- edited paragraph following Theorem 1.3, correcting a claim in the discussion of condition (i)