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THE DIVISOR FUNCTION IN ARITHMETIC PROGRESSIONS MODULO PRIME POWERS

Part of: Finite fields and commutative rings (number-theoretic aspects) Multiplicative number theory

Published online by Cambridge University Press:  17 May 2016

Rizwanur Khan
Rizwanur Khan*
Affiliation:
Science Program, Texas A&M University at Qatar, Doha, Qatar email rizwanur.khan@qatar.tamu.edu
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Abstract

We study the average value of the divisor function $\unicode[STIX]{x1D70F}(n)$ for $n\leqslant x$ with $n\equiv a~\text{mod}~q$ . The divisor function is known to be evenly distributed over arithmetic progressions for all $q$ that are a little smaller than $x^{2/3}$ . We show how to go past this barrier when $q=p^{k}$ for odd primes $p$ and any fixed integer $k\geqslant 7$ .

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 11T23: Exponential sums
Type
Research Article
Information
Mathematika , Volume 62 , Issue 3 , 2016 , pp. 898 - 908
Copyright
Copyright © University College London 2016 

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