Excursus: modular forms and more congruences for the partition function

2011 ◽  
pp. 195-203
Keyword(s):  
Partition Function ◽  
Modular Forms

Congruences for the Coefficients of Modular forms and Some New Congruences for the Partition Function

1957 ◽  
Vol 9 ◽  
pp. 549-552 ◽  
Author(s):  
Morris Newman
Keyword(s):  
Partition Function ◽  
Modular Forms ◽  
Negative Integer ◽  
Image Position

If n is a non-negative integer, define pr(n) as the coefficient of xn in;otherwise define pr(n) as 0. In a recent paper (2) the author established the following congruence:Let r = 4, 6, 8, 10, 14, 26. Let p be a prime greater than 3 such that r(p + l) / 24 is an integer, and set Δ = r(p2 − l)/24.

scholarly journals Rademacher expansions and the spectrum of 2d CFT

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Luis F. Alday ◽  
Jin-Beom Bae
Keyword(s):  
Partition Function ◽  
Modular Forms ◽  
Analytic Number Theory ◽  
Exact Formula ◽  
Conformal Field Theories ◽  
Light Spectrum ◽  
Chiral Algebra ◽  
Positive Weight ◽  
Conformal Field ◽  
Pure Gravity

Abstract A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and c > 1. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin j ≠ 0. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity.

scholarly journals Non-Existence of Ramanujan Congruences in Modular Forms of Level Four

2011 ◽  
Vol 63 (6) ◽  
pp. 1284-1306 ◽  
Author(s):  
Michael Dewar
Keyword(s):  
Partition Function ◽  
Modular Form ◽  
Half Plane ◽  
Modular Forms ◽  
Generating Functions ◽  
Open Questions ◽  
Ramanujan Congruences ◽  
Upper Half Plane

AbstractRamanujan famously found congruences like p(5n+4) ≡ 0 mod 5 for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Г1(4) that is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored F-partitions.

scholarly journals Asymptotic distribution of odd balanced unimodal sequences with rank congruent to a modulo c

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Taylor Garnowski
Keyword(s):  
Partition Function ◽  
Modular Form ◽  
Asymptotic Distribution ◽  
Generating Function ◽  
Modular Forms ◽  
Asymptotic Estimate ◽  
Asymptotic Properties ◽  
Unimodal Sequences ◽  
The World ◽  
Mock Modular Form

AbstractKim et al. (Proc Am Math Soc 144:687–3700, 2016) introduced the notion of odd-balance unimodal sequences in 2016. Like was shown by Bryson et al. (Proc Natl Acad Sci USA 109:16063–16067, 2012) for the generating function of strongly unimodal sequences, the generating function for odd-balanced unimodal sequences also has quantum modular behavior. Odd-balanced unimodal sequences thus appear to be a fundamental piece in the world of modular forms and combinatorics, and understanding their asymptotic properties is important for understanding their place in this puzzle. In light of this, we compute an asymptotic estimate for odd balanced unimodal sequences for ranks congruent to $$a \pmod {c}$$ a ( mod c ) for $$c\ne 2$$ c ≠ 2 or a multiple of 4. We find the interesting result that the odd balanced unimodal sequences are asymptotically related to the overpartition function. This is in contrast to strongly unimodal sequences which, are asymptotically related to the partition function. Our proofs of the main theorems rely on the representation of the generating function in question as a mixed mock modular form.

scholarly journals CONGRUENCES FOR THE COEFFICIENTS OF WEAKLY HOLOMORPHIC MODULAR FORMS

2006 ◽  
Vol 93 (2) ◽  
pp. 304-324 ◽  
Author(s):  
STEPHANIE TRENEER
Keyword(s):  
Partition Function ◽  
Wide Class ◽  
Modular Forms ◽  
Congruence Subgroup ◽  
Partition Functions ◽  
Modular Functions ◽  
Congruence Subgroups ◽  
Weakly Holomorphic Modular Forms ◽  
Linear Congruences ◽  
Holomorphic Modular Forms

Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomenon is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form on any congruence subgroup $\Gamma_0 (N)$. In particular, we give congruences for a wide class of partition functions and for traces of CM values of arbitrary modular functions on certain congruence subgroups of prime level.

THE RESTRICTED 3-COLORED PARTITION FUNCTION MOD 3

2013 ◽  
Vol 09 (07) ◽  
pp. 1789-1799
Author(s):  
BERNARD L. S. LIN
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Modular Forms ◽  
Type Identity ◽  
Colored Partitions ◽  
Combinatorial Interpretations

In this paper, we investigate the divisibility of the function b(n), counting the number of certain restricted 3-colored partitions of n. We obtain one Ramanujan type identity, which implies that b(3n + 2) ≡ 0 ( mod 3). Furthermore, we study the generating function for b(3n + 1) by modular forms. Finally, we find two cranks as combinatorial interpretations of the fact that b(3n + 2) is divisible by 3 for any n.

Hecke operators on certain subspaces of integral weight modular forms

2014 ◽  
Vol 10 (07) ◽  
pp. 1909-1919 ◽  
Author(s):  
Matthew Boylan ◽  
Kenny Brown
Keyword(s):  
Partition Function ◽  
Modular Forms ◽  
Invariant Subspaces ◽  
Hecke Operators ◽  
Dedekind Eta Function ◽  
Nagoya Math ◽  
Half Integral Weight ◽  
Fixed Weight ◽  
Integral Weight

Recent works of F. G. Garvan ([Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank, Int. J. Number Theory6(12) (2010) 281–309; MR2646759 (2011j:05032)]) and Y. Yang ([Congruences of the partition function, Int. Math. Res. Not.2011(14) (2011) 3261–3288; MR2817679 (2012e:11177)] and [Modular forms for half-integral weights on SL 2(ℤ), to appear in Nagoya Math. J.]) concern a certain family of half-integral weight Hecke-invariant subspaces which arise as multiples of fixed odd powers of the Dedekind eta-function multiplied by SL 2(ℤ)-forms of fixed weight. In this paper, we study the image of Hecke operators on subspaces which arise as multiples of fixed even powers of eta multiplied by SL 2(ℤ)-forms of fixed weight.

scholarly journals Mock theta functions and weakly holomorphic modular forms modulo 2 and 3

2014 ◽  
Vol 158 (1) ◽  
pp. 111-129 ◽  
Author(s):  
SCOTT AHLGREN ◽  
BYUNGCHAN KIM
Keyword(s):  
Partition Function ◽  
Wide Class ◽  
Modular Forms ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
Weakly Holomorphic Modular Forms ◽  
Linear Congruences ◽  
Image Position ◽  
Holomorphic Modular Forms

AbstractWe prove that the coefficients of the mock theta functions \begin{eqnarray*} f(q) = \sum_{n=1}^{\infty} \frac{ q^{n^2}}{(1+q)^2 (1+q^2)^2 \cdots (1+q^n)^2 } \end{eqnarray*} and \begin{eqnarray*} \omega(q)=1+\sum_{n=1}^\infty \frac{q^{2n^2+2n}}{(1+q)^2(1+q^3)^2\cdots (1+q^{2n+1})^2} \end{eqnarray*} possess no linear congruences modulo 3. We prove similar results for the moduli 2 and 3 for a wide class of weakly holomorphic modular forms and discuss applications. This extends work of Radu on the behavior of the ordinary partition function.

Sign in / Sign up

Export Citation Format

Share Document