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Averages of shifted convolutions of d3(n)

Part of: Additive number theory; partitions Multiplicative number theory Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  12 April 2012

S. Baier ,
T. D. Browning ,
G. Marasingha  and
L. Zhao
S. Baier
Affiliation:
Mathematisches Institut, Universität Göttingen, Bunsenstrasse 3–5, 37073 Göttingen, Germany (sbaier@uni-math.gwdg.de)
T. D. Browning
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK (t.d.browning@bristol.ac.uk; gihan.marasingha@bristol.ac.uk)
G. Marasingha
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK (t.d.browning@bristol.ac.uk; gihan.marasingha@bristol.ac.uk)
L. Zhao
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Republic of Singapore (lzhao@pmail.ntu.edu.sg)
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Abstract

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We investigate the first and second moments of shifted convolutions of the generalized divisor function d3(n).

Keywords

divisor functionconvolution sumsmoments of Riemann zeta functionHardy–Littlewood circle method

MSC classification

Secondary: 11N37: Asymptotic results on arithmetic functions 11M06: $zeta (s)$ and $L(s, chi)$ 11P55: Applications of the Hardy-Littlewood method
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 3 , October 2012 , pp. 551 - 576
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Carmichael, R., Expansions of arithmetical functions in infinite series, Proc. Lond. Math. Soc. 34 (1932), 1–26.CrossRefGoogle Scholar
2.Conrey, J. B. and Gonek, S. M., High moments of the Riemann zeta-function, Duke Math. J. 107 (2001), 577–604.Google Scholar
3.Duke, W., Friedlander, J. B. and Iwaniec, H., A quadratic divisor problem, Invent. Math. 115 (1994), 209–217.CrossRefGoogle Scholar
4.Hardy, G. H. and Littlewood, J. E., Contributions to the theory of the Riemann zetafunction and the theory of the distribution of primes, Acta Math. 41 (1918), 119–196.Google Scholar
5.Ingham, A. E., Mean-value theorems in the theory of the Riemann zeta-function, Proc. Lond. Math. Soc. 27 (1926), 273–300.Google Scholar
6.Ingham, A. E., Some asymptotic formulae in the theory of numbers, J. Lond. Math. Soc. 2 (1927), 202–208.Google Scholar
7.Ivić, A., The Riemann zeta-function(Wiley, Chichester, 1985).Google Scholar
8.Ivić, A., On the ternary additive divisor problem and the sixth moment of the Riemann zeta-function, in Sieve methods, exponential sums, and their applications in number theory, pp. 205–243 (Cambridge University Press, 1996).Google Scholar
9.Ivić, A., The general additive divisor problem and moments of the zeta-function, in Analytic and probabilistic methods in number theory, New Trends in Probability and Statistics, Volume 4, pp. 69–89 (VSP, Utrecht, 1997).Google Scholar
10.Iwaniec, H. and Kowalski, E., Analytic number theory, Colloquium Publications, Volume 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
11.Keating, J. P. and Snaith, N. C., Random matrix theory and ζ(1/2 + it), Commun. Math. Phys. 214 (2000) 57–89.CrossRefGoogle Scholar
12.Meurman, T., A generalization of Atkinson's formula to L-functions, Acta Arith. 47 (1986), 351–370.Google Scholar
13.Meurman, T., On the binary additive divisor problem, in Number theory, De Gruyter Proceedings in Mathematics, pp. 223–246 (Walter de Gruyter, Berlin, 2001).Google Scholar
14.Mikawa, H., On prime twins, Tsukuba J. Math. 15 (1991), 19–29.Google Scholar
15.Motohashi, Y., The binary additive divisor problem, Annales Scient. Éc. Norm. Sup. 27 (1994), 529–572.Google Scholar
16.Titchmarsh, E. C., The theory of the Riemann zeta-function, 2nd edn (Oxford University Press, 1986).Google Scholar