Gaps between nonzero Fourier coefficients of cusp forms

Das, Soumya; Ganguly, Satadal

doi:10.1090/S0002-9939-2014-12164-2

Gaps between nonzero Fourier coefficients of cusp forms
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by Soumya Das and Satadal Ganguly

Proc. Amer. Math. Soc. 142 (2014), 3747-3755

DOI: https://doi.org/10.1090/S0002-9939-2014-12164-2

Published electronically: July 24, 2014

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Abstract:

We prove that for any even integer $k \geq 12$, there are positive constants $c$ and $X_0$ that depend only on $k$ such that for all nonzero cusp forms $f$ of weight $k$ for the full modular group, any interval $(X, X+c X^{1/4})$ with $X>X_0$ must contain an integer $n$ with the $n$-th Fourier coefficient of $f$ nonzero.

References

Emre Alkan, Nonvanishing of Fourier coefficients of modular forms, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1673–1680. MR 1953571, DOI 10.1090/S0002-9939-02-06758-8
Emre Alkan, On the sizes of gaps in the Fourier expansion of modular forms, Canad. J. Math. 57 (2005), no. 3, 449–470. MR 2134398, DOI 10.4153/CJM-2005-019-7
Emre Alkan and Alexandru Zaharescu, Nonvanishing of the Ramanujan tau function in short intervals, Int. J. Number Theory 1 (2005), no. 1, 45–51. MR 2172330, DOI 10.1142/S1793042105000029
Emre Alkan and Alexandru Zaharescu, Nonvanishing of Fourier coefficients of newforms in progressions, Acta Arith. 116 (2005), no. 1, 81–98. MR 2114906, DOI 10.4064/aa116-1-7
Emre Alkan and Alexandru Zaharescu, On the gaps in the Fourier expansion of cusp forms, Ramanujan J. 16 (2008), no. 1, 41–52. MR 2407238, DOI 10.1007/s11139-007-9091-z
Antal Balog and Ken Ono, The Chebotarev density theorem in short intervals and some questions of Serre, J. Number Theory 91 (2001), no. 2, 356–371. MR 1876282, DOI 10.1006/jnth.2001.2694
R. P. Bambah and S. Chowla, On numbers which can be expressed as a sum of two squares, Proc. Nat. Inst. Sci. India 13 (1947), 101–103. MR 22879
Kazuyuki Hatada, Eigenvalues of Hecke operators on $\textrm {SL}(2,\,\textbf {Z})$, Math. Ann. 239 (1979), no. 1, 75–96. MR 516060, DOI 10.1007/BF01420494
M. N. Huxley, Exponential sums and lattice points. III, Proc. London Math. Soc. (3) 87 (2003), no. 3, 591–609. MR 2005876, DOI 10.1112/S0024611503014485
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
Winfried Hohnen, Period polynomials and congruences for Hecke algebras [eigenvalues], Proc. Edinburgh Math. Soc. (2) 42 (1999), no. 2, 217–224. MR 1697394, DOI 10.1017/S0013091500020204
Emmanuel Kowalski, Olivier Robert, and Jie Wu, Small gaps in coefficients of $L$-functions and $\mathfrak {B}$-free numbers in short intervals, Rev. Mat. Iberoam. 23 (2007), no. 1, 281–326. MR 2351136, DOI 10.4171/RMI/496
Ekkehard Krätzel, Squarefree numbers as sums of two squares, Arch. Math. (Basel) 39 (1982), no. 1, 28–31. MR 674530, DOI 10.1007/BF01899241
D. H. Lehmer, The vanishing of Ramanujan’s function $\tau (n)$, Duke Math. J. 14 (1947), 429–433. MR 21027
M. Ram Murty and V. Kumar Murty, Odd values of Fourier coefficients of certain modular forms, Int. J. Number Theory 3 (2007), no. 3, 455–470. MR 2352830, DOI 10.1142/S1793042107001036
Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
Wenguang Zhai, Square-free integers as sums of two squares, Number theory, Dev. Math., vol. 15, Springer, New York, 2006, pp. 219–227. MR 2213838, DOI 10.1007/0-387-30829-6_{1}5