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ON THE EXPONENT OF DISTRIBUTION OF THE TERNARY DIVISOR FUNCTION

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

Published online by Cambridge University Press:  02 June 2014

Étienne Fouvry ,
Emmanuel Kowalski  and
Philippe Michel
Étienne Fouvry
Affiliation:
Université Paris Sud, Laboratoire de Mathématique, Campus d’Orsay, 91405 Orsay Cedex, France email etienne.fouvry@math.u-psud.fr
Emmanuel Kowalski
Affiliation:
ETH Zürich – D-MATH, Rämistrasse 101, CH-8092 Zürich, Switzerland email kowalski@math.ethz.ch
Philippe Michel
Affiliation:
EPFL/SB/IMB/TAN, Station 8, CH-1015 Lausanne, Switzerland email philippe.michel@epfl.ch
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Abstract

We show that the exponent of distribution of the ternary divisor function $d_{3}$ in arithmetic progressions to prime moduli is at least $1/2+1/46$, improving results of Friedlander–Iwaniec and Heath-Brown. Furthermore, when averaging over a fixed residue class, we prove that this exponent is increased to $1/2+1/34$.

MSC classification

Primary: 11L05: Gauss and Kloosterman sums; generalizations 11N25: Distribution of integers with specified multiplicative constraints 11N37: Asymptotic results on arithmetic functions 11T23: Exponential sums
Type
Research Article
Information
Mathematika , Volume 61 , Issue 1 , January 2015 , pp. 121 - 144
Copyright
Copyright © University College London 2014 

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