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THE DIVISOR FUNCTION IN ARITHMETIC PROGRESSIONS MODULO PRIME POWERS

Published online by Cambridge University Press:  17 May 2016

Rizwanur Khan*
Affiliation:
Science Program, Texas A&M University at Qatar, Doha, Qatar email [email protected]
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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 $k\geqslant 7$ .

Type
Research Article
Copyright
Copyright © University College London 2016 

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