Sato–Tate equidistribution for families of
Hecke–Maass forms on SL(n, ℝ)∕SO(n)

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

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EXTREME VALUES OF GEODESIC PERIODS ON ARITHMETIC HYPERBOLIC SURFACES

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s147474802000064x ◽

2020 ◽

pp. 1-36

Author(s):

Bart Michels

Keyword(s):

Lower Bound ◽

Closed Geodesic ◽

Extreme Values ◽

Theta Series ◽

Hyperbolic Surface ◽

Quadratic Fields ◽

Maass Forms ◽

Real Quadratic Fields ◽

Central Values ◽

Real Quadratic

Abstract Given a closed geodesic on a compact arithmetic hyperbolic surface, we show the existence of a sequence of Laplacian eigenfunctions whose integrals along the geodesic exhibit nontrivial growth. Via Waldspurger’s formula we deduce a lower bound for central values of Rankin-Selberg L-functions of Maass forms times theta series associated to real quadratic fields.

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

International Journal of Number Theory ◽

10.1142/s179304211000296x ◽

2010 ◽

Vol 06 (02) ◽

pp. 281-309 ◽

Cited By ~ 32

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.

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Twisted traces of CM values of weak Maass forms

Journal of Number Theory ◽

10.1016/j.jnt.2012.10.008 ◽

2013 ◽

Vol 133 (6) ◽

pp. 1827-1845 ◽

Cited By ~ 11

Author(s):

Claudia Alfes ◽

Stephan Ehlen

Keyword(s):

Maass Forms

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On effective determination of symmetric-square lifts

Open Mathematics ◽

10.2478/s11533-014-0404-3 ◽

2014 ◽

Vol 12 (7) ◽

Cited By ~ 1

Author(s):

Qingfeng Sun

Keyword(s):

Cusp Forms ◽

Maass Forms ◽

Central Values ◽

Laplace Eigenvalue ◽

Symmetric Square ◽

Ramanujan Conjecture

AbstractLet F be the symmetric-square lift with Laplace eigenvalue λ F (Δ) = 1+4µ2. Suppose that |µ| ≤ Λ. We show that F is uniquely determined by the central values of Rankin-Selberg L-functions L(s, F ⋇ h), where h runs over the set of holomorphic Hecke eigen cusp forms of weight κ ≡ 0 (mod 4) with κ≍ϱ+ɛ, t9 = max {4(1+4θ)/(1−18θ), 8(2−9θ)/3(1−18θ)} for any 0 ≤ θ < 1/18 and any ∈ > 0. Here θ is the exponent towards the Ramanujan conjecture for GL2 Maass forms.

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On exponential sums for coefficients of general L-functions

International Journal of Number Theory ◽

10.1142/s1793042121500664 ◽

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].

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