Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-02-11T22:48:36.363Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE DISTRIBUTION OF -FREE NUMBERS AND NON-VANISHING FOURIER COEFFICIENTS OF CUSP FORMS

Published online by Cambridge University Press:  29 March 2012

KAISA MATOMÄKI
KAISA MATOMÄKI*
Affiliation:
Department of Mathematics, University of Turku, 20014 Turku, Finland e-mail: ksmato@utu.fi
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study properties of -free numbers, that is numbers that are not divisible by any member of a set . First we formulate the most-used procedure for finding them (in a given set of integers) as easy-to-apply propositions. Then we use the propositions to consider Diophantine properties of -free numbers and their distribution on almost all short intervals. Results on -free numbers have implications to non-vanishing Fourier coefficients of cusp forms, so this work also gives information about them.

Keywords

11F3011J2511N25
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 2 , May 2012 , pp. 381 - 397
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Alkan, E., Nonvanishing of Fourier coefficients of modular forms, Proc. Amer. Math. Soc. 131 (2003), 1673–1680.CrossRefGoogle Scholar
2.Alkan, E., On the sizes of gaps in the Fourier expansion of modular forms, Canad. J. Math. 57 (2005), 449–470.CrossRefGoogle Scholar
3.Alkan, E., Average size of gaps in the Fourier expansion of modular forms, Int. J. Number Theory. 3 (2007), 207–215.CrossRefGoogle Scholar
4.Alkan, E., Harman, G. and Zaharescu, A.. Diophantine approximation with mild divisibility constraints. J. Number Theory. 118 (2006), 1–14.CrossRefGoogle Scholar
5.Alkan, E. and Zaharescu, A., B-free numbers in short arithmetic progressions, J. Number Theory. 113 (2005), 226–243.CrossRefGoogle Scholar
6.Baker, R. C., Diophantine inequalities, volume 1 of London Mathematical Society Monographs (New Series) (Clarendon Press, Oxford, 1986).Google Scholar
7.Balog, A. and Ono, K., The Chebotarev density theorem in short intervals and some questions of Serre, J. Number Theory. 91 (2001), 356–371.CrossRefGoogle Scholar
8.Deshouillers, J. M. and Iwaniec, H., Power mean-values for Dirichlet's polynomials and the Riemann zeta-function. II, Acta. Arith. 43 (1983), 305–312.CrossRefGoogle Scholar
9.Erdős, P., On the difference of consecutive terms of sequences defined by divisibility properties, Acta. Arith. 12 (1966), 175–182.CrossRefGoogle Scholar
10.Friedlander, J. and Iwaniec, H., On Bombieri's asymptotic sieve, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 719–756.Google Scholar
11.Harman, G., Prime-detecting sieves, volume 33 of London Mathematical Society Monographs (New Series) (Princeton University Press, Princeton, 2007).Google Scholar
12.Iwaniec, H. and Kowalski, E.. Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications (American Mathematical Society, Providence, Rhode Island, 2004).Google Scholar
13.Kowalski, E., Robert, O. and Wu, J., Small gaps in coefficients of L-functions and -free numbers in short intervals, Rev. Mat. Iberoam. 23 (2007) 281–326.CrossRefGoogle Scholar
14.Matomäki, K., The distribution of αp modulo one, Math. Proc. Cambridge Philos. Soc. 147 (2009) 267–283.CrossRefGoogle Scholar
15.Peck, A. S., On the differences between consecutive primes. PhD Thesis (University of Oxford, 1996).Google Scholar
16.Plaksin, V. A., Distribution of -free numbers, Mat. Zametki. 47 (1990), 69–77.Google Scholar
17.Serre, J.-P., Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401.CrossRefGoogle Scholar
18.Watt, N., Kloosterman sums and a mean value for Dirichlet polynomials, J. Number Theory. 53 (1995) 179–210.CrossRefGoogle Scholar
19.Wu, J., Distribution des nombres -libres dans de petits intervalles, J. Théor. Nombres Bordeaux. 5 (1993) 151–163.CrossRefGoogle Scholar
20.Wu, J., Nombres -libres dans les petits intervalles. Acta. Arith. 65 (2) (1993), 97–116.CrossRefGoogle Scholar