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 \(\log {x}\).
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Notes
That is, the limit relation (1.2) holds uniformly in q, for these q.
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Acknowledgements
We thank the referee for carefully reading the manuscript and for making helpful suggestions that have improved the results and the exposition. The first named author (P.P.) is supported by NSF award DMS-2001581.
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Pollack, P., Singha Roy, A. Distribution in coprime residue classes of polynomially-defined multiplicative functions. Math. Z. 303, 93 (2023). https://doi.org/10.1007/s00209-023-03240-7
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DOI: https://doi.org/10.1007/s00209-023-03240-7
Keywords
- Uniform distribution
- Equidistribution
- Weak uniform distribution
- Weak equidistribution
- Multiplicative function