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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
Abstract
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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
Secondary:
11T23: Exponential sums
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- Copyright © University College London 2016
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