scholarly journals ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS

2010 ◽  
Vol 06 (01) ◽  
pp. 185-202 ◽  
Author(s):  
MATTHEW BOYLAN
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Hecke Operators ◽  
Mock Theta Function ◽  
Maass Forms ◽  
Maass Form ◽  
Arithmetic Properties ◽  
Holomorphic Projection ◽  
Harmonic Weak Maass Form ◽  
Integer Weight

In a recent work, Bringmann and Ono [4] show that Ramanujan's f(q) mock theta function is the holomorphic projection of a harmonic weak Maass form of weight 1/2. In this paper, we extend the work of Ono in [13]. In particular, we study holomorphic projections of certain integer weight harmonic weak Maass forms on SL 2(ℤ) using Hecke operators and the differential theta-operator.

scholarly journals Higher depth mock theta functions and q-hypergeometric series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joshua Males ◽  
Andreas Mono ◽  
Larry Rolen
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Mock Modular Forms ◽  
Harmonic Maass Forms

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.

THE COEFFICIENTS OF THE ω(q) MOCK THETA FUNCTION

2008 ◽  
Vol 04 (06) ◽  
pp. 1027-1042 ◽  
Author(s):  
SHARON ANNE GARTHWAITE
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Theta Functions ◽  
Exact Formula ◽  
Poincaré Series ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Poincare Series ◽  
Vector Valued

In 1920, Ramanujan wrote to Hardy about his discovery of the mock theta functions. In the years since, there has been much work in understanding the transformation properties and asymptotic nature of these functions. Recently, Zwegers proved a relationship between mock theta functions and vector-valued modular forms, and Bringmann and Ono used the theory of Maass forms and Poincaré series to prove a conjecture of Andrews, yielding an exact formula for the coefficients of the f(q) mock theta function. Here we build upon these results, using the theory of vector-valued modular forms and Poincaré series to prove an exact formula for the coefficients of the ω(q) mock theta function.

scholarly journals Jacobi’s triple product, mock theta functions, unimodal sequences and the q-bracket

2018 ◽  
Vol 14 (07) ◽  
pp. 1961-1981
Author(s):  
Robert Schneider
Keyword(s):  
Unit Circle ◽  
Theta Function ◽  
Modular Forms ◽  
Statistical Physics ◽  
Theta Functions ◽  
Triple Product ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Quantum Modular Forms ◽  
Unimodal Sequences

In Ramanujan’s final letter to Hardy, he listed examples of a strange new class of infinite series he called “mock theta functions”. It turns out all of these examples are essentially specializations of a so-called universal mock theta function [Formula: see text] of Gordon–McIntosh. Here we show that [Formula: see text] arises naturally from the reciprocal of the classical Jacobi triple product—and is intimately tied to rank generating functions for unimodal sequences, which are connected to mock modular and quantum modular forms—under the action of an operator related to statistical physics and partition theory, the [Formula: see text]-bracket of Bloch–Okounkov. Second, we find [Formula: see text] to extend in [Formula: see text] to the entire complex plane minus the unit circle, and give a finite formula for this universal mock theta function at roots of unity, that is simple by comparison to other such formulas in the literature; we also indicate similar formulas for other [Formula: see text]-hypergeometric series. Finally, we look at interesting “quantum” behaviors of mock theta functions inside, outside, and on the unit circle.

scholarly journals Maass forms and the mock theta function f(q)

2019 ◽  
Vol 374 (3-4) ◽  
pp. 1681-1718
Author(s):  
Scott Ahlgren ◽  
Alexander Dunn
Keyword(s):  
Theta Function ◽  
Mock Theta Function ◽  
Maass Forms

ARITHMETIC PROPERTIES OF COEFFICIENTS OF THE MOCK THETA FUNCTION

2019 ◽  
Vol 102 (1) ◽  
pp. 50-58
Author(s):  
RENRONG MAO
Keyword(s):  
Theta Function ◽  
Second Order ◽  
Arithmetic Progressions ◽  
Mock Theta Function ◽  
Arithmetic Properties ◽  
Order Mock Theta Function

We investigate the arithmetic properties of the second-order mock theta function $B(q)$ and establish two identities for the coefficients of this function along arithmetic progressions. As applications, we prove several congruences for these coefficients.

scholarly journals On second order mock theta function $ B(q) $

2021 ◽  
Vol 30 (1) ◽  
pp. 52-65
Author(s):  
Harman Kaur ◽  
◽  
Meenakshi Rana
Keyword(s):  
Theta Function ◽  
Second Order ◽  
Mock Theta Function ◽  
Arithmetic Properties ◽  
Order Mock Theta Function

<abstract><p>In this paper, we present some arithmetic properties for the second order mock theta function $ B(q) $ given by McIntosh as:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ B(q) = \sum\limits_{n = 0}^{\infty}\frac{q^n(-q;q^2)_n}{(q;q^2)_{n+1}}. $\end{document} </tex-math></disp-formula></p> </abstract>

scholarly journals MOCK THETA FUNCTIONS AND QUANTUM MODULAR FORMS

2013 ◽  
Vol 1 ◽  
Author(s):  
AMANDA FOLSOM ◽  
KEN ONO ◽  
ROBERT C. RHOADES
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Asymptotic Properties ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Radial Limits ◽  
Quantum Modular Forms ◽  
Even Order ◽  
Image Position

AbstractRamanujan’s last letter to Hardy concerns the asymptotic properties of modular forms and his ‘mock theta functions’. For the mock theta function $f(q)$, Ramanujan claims that as $q$ approaches an even-order $2k$ root of unity, we have $$\begin{eqnarray*}f(q)- (- 1)^{k} (1- q)(1- {q}^{3} )(1- {q}^{5} )\cdots (1- 2q+ 2{q}^{4} - \cdots )= O(1).\end{eqnarray*}$$ We prove Ramanujan’s claim as a special case of a more general result. The implied constants in Ramanujan’s claim are not mysterious. They arise in Zagier’s theory of ‘quantum modular forms’. We provide explicit closed expressions for these ‘radial limits’ as values of a ‘quantum’ $q$-hypergeometric function which underlies a new relationship between Dyson’s rank mock theta function and the Andrews–Garvan crank modular form. Along these lines, we show that the Rogers–Fine false $\vartheta $-functions, functions which have not been well understood within the theory of modular forms, specialize to quantum modular forms.

scholarly journals On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

2019 ◽  
Vol 17 (1) ◽  
pp. 1631-1651
Author(s):  
Ick Sun Eum ◽  
Ho Yun Jung
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Congruence Subgroups ◽  
Maass Form ◽  
Weakly Holomorphic Modular Forms ◽  
Reciprocity Law ◽  
Class Invariants ◽  
Higher Genus ◽  
Holomorphic Modular Forms

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

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.

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