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ON THE EXPONENT OF DISTRIBUTION OF THE TERNARY DIVISOR FUNCTION

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

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 [email protected]
Emmanuel Kowalski
Affiliation:
ETH Zürich – D-MATH, Rämistrasse 101, CH-8092 Zürich, Switzerland email [email protected]
Philippe Michel
Affiliation:
EPFL/SB/IMB/TAN, Station 8, CH-1015 Lausanne, Switzerland email [email protected]
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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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References

Bombieri, E., Friedlander, J. and Iwaniec, H., Primes in arithmetic progressions to large moduli. Acta Math. 156 1985, 203–251.Google Scholar
Bombieri, E., Friedlander, J. and Iwaniec, H., Primes in arithmetic progressions to large moduli. II. Math. Ann. 277(3) 1987, 361–393.Google Scholar
Deshouillers, J.-M. and Iwaniec, H., Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70 1982/1983, 219–288.Google Scholar
Fouvry, É., Autour du théorème de Bombieri–Vinogradov. Acta Math. 152(3–4) 1984, 219–244.Google Scholar
Fouvry, É., Sur le problème des diviseurs de Titchmarsh. J. Reine Angew. Math. 357 1985, 51–76.Google Scholar
Fouvry, É. and Iwaniec, H., Primes in arithmetic progressions. Acta Arith. 42(2) 1983, 197–218.CrossRefGoogle Scholar
Fouvry, É., Michel, Ph. and Kowalski, E., Algebraic twists of modular forms and Hecke orbits. Preprint, 2012, arXiv:1207.0617.Google Scholar
Fouvry, É., Michel, Ph. and Kowalski, E., Algebraic trace functions over the primes. Duke Math. J. (to appear); arXiv:1211.6043.Google Scholar
Friedlander, J. B. and Iwaniec, H., Incomplete Kloosterman sums and a divisor problem (with an appendix by B. J. Birch and E. Bombieri). Ann. of Math. (2) 121(2) 1985, 319–350.CrossRefGoogle Scholar
Heath-Brown, D. R., The divisor function d 3(n) in arithmetic progressions. Acta Arith. 47 1986, 29–56.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic Number Theory (American Mathematical Society Colloquium Publications 53), American Mathematical Society (Providence, RI, 2004).Google Scholar
Motohashi, Y., An induction principle for the generalization of Bombieri’s prime number theorem. Proc. Japan Acad. 52(6) 1976, 273–275.Google Scholar
Polymath, D. H. J., New equidistribution estimates of Zhang type, and bounded gaps between primes. Preprint, 2014, arXiv:1402.0811.Google Scholar
Shiu, P., A Brun-Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313 1980, 161–170.Google Scholar
Wolke, D., Über die mittlere Verteilung der Werte zahlentheoretischer Funktionen auf Restklassen, I. Math. Ann. 202 1973, 1–25.Google Scholar
Zhang, Y., Bounded gaps between primes. Ann. of Math. (2) 179(3) 2014, 1121–1174; doi: 10.4007/annals.2014.179.3.7.CrossRefGoogle Scholar