Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-15T22:23:24.375Z Has data issue: false hasContentIssue false
  • English
  • Français

Nair–Tenenbaum bounds uniform with respect to the discriminant

Published online by Cambridge University Press:  12 January 2012

KEVIN HENRIOT
KEVIN HENRIOT*
Affiliation:
Université de Montréal, Département de Mathématiques et de Statistique, Pavillon André–Aisenstadt, Bureau 5190, 2900 Édouard–Montpetit, Montréal, Québec, Canada, H3C 3J7. e-mail: henriot@dms.umontreal.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

For functions F satisfying a certain submultiplicativity condition and polynomials Q1, . . ., Qk in [X], Nair and Tenenbaum obtained an upper bound on the short sum with an implicit dependency on the discriminant of Q1 . . . Qk. We obtain a similar upper bound uniform in the discriminant.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 3 , May 2012 , pp. 405 - 424
Copyright
Copyright © Cambridge Philosophical Society 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]de la Bretèche, R. and Browning, T. D.Sums of arithmetic functions over values of binary forms. Acta Arith. 125 (2007), 291–304.CrossRefGoogle Scholar
[2]de la Bretèche, R., Browning, T. D. and Peyre, E. On Manin's conjecture for a family of Chatelet surfaces. To appear in Ann. of Math. (2010).Google Scholar
[3]de la Bretèche, R. and Tenenbaum, G. Moyenne de fonctions arithmétiques de formes binaires. To appear in Mathematika (2011).CrossRefGoogle Scholar
[4]Daniel, S. Uniform bounds for short sums of certain arithmetic functions of polynomial arguments. Unpublished manuscript.Google Scholar
[5]Erdös, P.On the sum ∑k=1xd(f(k)). J. London Math. Soc. 27 (1952), 7–15.CrossRefGoogle Scholar
[6]Halberstam, H. and Richert, H.-E.Sieve Methods (Academic Press, 1974).Google Scholar
[7]Holowinsky, R.Sieving for Mass equidistribution. Ann. of Math. (2) 172 (2010), 1499–1516.CrossRefGoogle Scholar
[8]Holowinsky, R. and Soundararajan, K.Mass equidistribution for Hecke eigenforms. Ann. of Math. (2) 172 (2010), 1517–1528.CrossRefGoogle Scholar
[9]Lang, S.Algebra (Spinger–Verlag, 2002).CrossRefGoogle Scholar
[10]Matthiesen, L. Correlations of the divisor function. Preprint. arXiv:1011.0019 (2010).Google Scholar
[11]Nagell, T.Introduction to Number Theory (Wiley, 1951).Google Scholar
[12]Nair, M.Multiplicative functions of polynomial values in short intervals. Acta Arith. 62 (1992), 257–269.CrossRefGoogle Scholar
[13]Nair, M. and Tenenbaum, G.Short sums of certain arithmetic functions. Acta Math. 180 (1998), 119–144.CrossRefGoogle Scholar
[14]Pritsker, I. E.An inequality for the norm of a polynomial factor. Proc. Amer. Math. Soc. 129 (2001), 2283–2291.CrossRefGoogle Scholar
[15]Shiu, P.A Brun–Titschmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313 (1980), 161–170.Google Scholar
[16]Stewart, C. L.On the number of solutions of polynomial congruences and Thue equations. J. Amer. Math. Soc. 4 (1991), 793–835.CrossRefGoogle Scholar