Fourier coefficients of harmonic weak Maass forms and the partition function

2015 ◽  
Vol 137 (4) ◽  
pp. 1061-1097 ◽  
Author(s):  
Riad Masri
Keyword(s):  
Partition Function ◽  
Fourier Coefficients ◽  
Maass Forms

scholarly journals CONGRUENCES FOR ANDREWS' SMALLEST PARTS PARTITION FUNCTION AND NEW CONGRUENCES FOR DYSON'S RANK

2010 ◽  
Vol 06 (02) ◽  
pp. 281-309 ◽  
Author(s):  
F. G. GARVAN
Keyword(s):  
Partition Function ◽  
Rank Function ◽  
Maass Forms ◽  
Hecke Eigenforms ◽  
Half Integer ◽  
Integer Weight

Let spt (n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt (n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt (n) mod ℓ for ℓ = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.

On exponential sums for coefficients of general L-functions

2021 ◽  
pp. 1-22
Author(s):  
Fei Hou
Keyword(s):  
Functional Equation ◽  
General Formula ◽  
Fourier Coefficients ◽  
Exponential Sums ◽  
Exponential Functions ◽  
Maass Forms ◽  
Maass Form ◽  
Rapid Decay

We investigate the order of exponential sums involving the coefficients of general [Formula: see text]-functions satisfying a suitable functional equation and give some new estimates, including refining certain results in preceding works [X. Ren and Y. Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for [Formula: see text], Sci. China Math. 58(10) (2015) 2105–2124; Y. Jiang and G. Lü, Oscillations of Fourier coefficients of Hecke–Maass forms and nonlinear exponential functions at primes, Funct. Approx. Comment. Math. 57 (2017) 185–204].

scholarly journals On the Fourier coefficients of meromorphic Jacobi forms

2014 ◽  
Vol 10 (06) ◽  
pp. 1519-1540 ◽  
Author(s):  
René Olivetto
Keyword(s):  
Analytic Functions ◽  
Fourier Coefficients ◽  
Jacobi Form ◽  
Jacobi Forms ◽  
Precise Description ◽  
Maass Forms ◽  
Real Analytic Functions ◽  
Harmonic Maass Forms ◽  
Real Analytic ◽  
Vector Valued

In this paper, we describe the automorphic properties of the Fourier coefficients of meromorphic Jacobi forms. Extending results of Dabholkar, Murthy, and Zagier, and Bringmann and Folsom, we prove that the canonical Fourier coefficients of a meromorphic Jacobi form φ(z; τ) are the holomorphic parts of some (vector-valued) almost harmonic Maass forms. We also give a precise description of their completions, which turn out to be uniquely determined by the Laurent coefficients of φ at each pole, as well as some well-known real analytic functions, that appear for instance in the completion of Appell–Lerch sums.

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