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Asymptotic formulas for the coefficients of certain automorphic functions. (English) Zbl 1370.11056
Summary: We derive asymptotic formulas for the coefficients of certain classes of weakly holomorphic Jacobi forms and weakly holomorphic modular forms (not necessarily of integral weight) without using the circle method. Then two applications of these formulas are given. Namely, we estimate the growth of the Fourier coefficients of two important weak Jacobi forms of index \(1\) and non-positive weights and obtain an asymptotic formula for the Fourier coefficients of the modular functions \(\theta ^k/\eta ^l\) for all integers \(k,l\geq 1\), where \(\theta \) is the weight \(1/2\) modular form and \(\eta \) is the Dedekind eta function.
11F30 Fourier coefficients of automorphic forms
11F50 Jacobi forms
11F03 Modular and automorphic functions
Full Text: DOI
[1] [1]K. Bringmann and O. Richter, Zagier-type dualities and lifting maps for harmonic Maass–Jacobi forms, Adv. Math. 225 (2010), 2298–2315. · Zbl 1264.11039
[2] [2]K. Bringmann and O. Richter, Exact formulas for coefficients of Jacobi forms, Int. J. Number Theory 7 (2011), 825–833. · Zbl 1279.11050
[3] [3]A. Dabholkar, S. Murthy and D. Zagier, Quantum black holes, wall crossing, and mock modular forms, arXiv:1208.4074v2 (2014); to appear in Cambridge Monogr. Math. Phys.
[4] [4]M. Dewar and M. R. Murty, A derivation of the Hardy–Ramanujan formula from an arithmetic formula, Proc. Amer. Math. Soc. 141 (2013), 1903–1911. · Zbl 1357.11104
[5] [5]M. Dewar and M. R. Murty, An asymptotic formula for the coefficients of j(z), Int. J. Number Theory 9 (2013), 641–652. · Zbl 1335.11033
[6] [6]M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progr. Math. 55, Birkh\"auser, 1985. 76J. Meher and K. Deo Shankhadhar · Zbl 0554.10018
[7] [7]G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. 17 (1918), 75–115. · JFM 46.0198.04
[8] [8]H. Petersson, \"Uber die Entwicklungskoeffizienten der automorphen Formen, Acta Math. 58 (1932), 169–215. · JFM 58.1110.01
[9] [9]H. Rademacher, The Fourier coefficients of the modular invariant j(\(\tau\) ), Amer. J. Math. 60 (1938), 501–512. · JFM 64.0122.01
[10] [10]H. Rademacher and H. S. Zuckerman, On the Fourier coefficients of certain modular forms of positive dimension, Ann. of Math. 39 (1938), 433–462. · Zbl 0019.02201
[11] [11]H. S. Zuckerman, On the coefficients of certain modular forms belonging to subgroups of the modular group, Trans. Amer. Math. Soc. 45 (1939), 298–321. · JFM 65.0352.01
[12] [12]H. S. Zuckerman, On the expansions of certain modular forms of positive dimension, Amer. J. Math. 62 (1940), 127–152. · JFM 66.0373.01
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