PARITY OF THE PARTITION FUNCTION AND TRACES OF SINGULAR MODULI

2012 ◽  
Vol 08 (02) ◽  
pp. 395-409 ◽  
Author(s):  
PAK-HIN LEE ◽  
ALEXANDR ZAMORZAEV
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Logarithmic Derivative ◽  
Complex Multiplication ◽  
Hecke Operators ◽  
Singular Moduli ◽  
Class Polynomials

We prove that the parity of the partition function is given by the "trace" of the Hauptmodul [Formula: see text] for [Formula: see text] at points of complex multiplication. Using Hecke operators, we generalize this to relate the Hecke traces of [Formula: see text] to the partition function modulo 2. We then prove that the generating function for these Hecke traces is equal to the logarithmic derivative of the level 6 Hilbert class polynomial. Finally, we give a procedure involving Hilbert class polynomials for computing the parity of the partition function, and make some speculations about the distribution of these universal polynomials modulo class polynomials.

scholarly journals HECKE-TYPE CONGRUENCES FOR ANDREWS' SPT-FUNCTION MODULO 16 AND 32

2014 ◽  
Vol 10 (02) ◽  
pp. 375-390 ◽  
Author(s):  
FRANK G. GARVAN ◽  
CHRIS JENNINGS-SHAFFER
Keyword(s):  
Generating Function ◽  
Hecke Operators ◽  
Powers Of 2

Inspired by recent congruences by Andersen with varying powers of 2 in the modulus for partition related functions, we extend the modulo 32760 congruences of the first author for the function spt (n). We show that a normalized form of the generating function of spt (n) is an eigenform modulo 32 for the Hecke operators T(ℓ2) for primes ℓ ≥ 5 with ℓ ≡ 1, 11, 17, 19 (mod 24), and an eigenform modulo 16 for ℓ ≡ 13, 23 (mod 24).

Ramanujan Congruences For p-k(n) Modulo Powers Of 17

1991 ◽  
Vol 43 (3) ◽  
pp. 506-525 ◽  
Author(s):  
Kim Hughes
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Ramanujan Congruences ◽  
Image Position

For each integer r we define the sequence pr(n) by We note that p-1(n) = p(n), the ordinary partition function. On account of this some authors set r = — k to make positive values of k correspond to positive powers of the generating function for p(n): We follow this convention here. In [3], Atkin proved the following theorem.

Inverse Problems for Partition Functions

2001 ◽  
Vol 53 (4) ◽  
pp. 866-896
Author(s):  
Yifan Yang
Keyword(s):  
Asymptotic Behavior ◽  
Partition Function ◽  
Inverse Problems ◽  
Large Class ◽  
Generating Function ◽  
Riemann Hypothesis ◽  
Arithmetic Function ◽  
Partition Functions ◽  
Class Of Functions ◽  
Weighted Partition

AbstractLet pw(n) be the weighted partition function defined by the generating function , where w(m) is a non-negative arithmetic function. Let be the summatory functions for pw(n) and w(n), respectively. Generalizing results of G. A. Freiman and E. E. Kohlbecker, we show that, for a large class of functions Φ(u) and λ(u), an estimate for Pw(u) of the formlog Pw(u) = Φ(u){1 + Ou(1/λ(u))} (u→∞) implies an estimate forNw(u) of the formNw(u) = Φ*(u){1+O(1/ log ƛ(u))} (u→∞) with a suitable function Φ*(u) defined in terms of Φ(u). We apply this result and related results to obtain characterizations of the Riemann Hypothesis and the Generalized Riemann Hypothesis in terms of the asymptotic behavior of certain weighted partition functions.

scholarly journals Brane webs and random processes

2015 ◽  
Vol 30 (33) ◽  
pp. 1550202 ◽  
Author(s):  
Amer Iqbal ◽  
Babar A. Qureshi ◽  
Khurram Shabbir ◽  
Muhammad A. Shehper
Keyword(s):  
Partition Function ◽  
Boundary Conditions ◽  
Generating Function ◽  
Random Processes ◽  
Open String ◽  
Plane Partitions ◽  
Schur Process ◽  
String Amplitudes

We study (p, q) 5-brane webs dual to certain N M5-brane configurations and show that the partition function of these brane webs gives rise to cylindric Schur process with period N. This generalizes the previously studied case of period 1. We also show that open string amplitudes corresponding to these brane webs are captured by the generating function of cylindric plane partitions with profile determined by the boundary conditions imposed on the open string amplitudes.

scholarly journals Overpartitions with Restricted Odd Differences

10.37236/5248 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Kathrin Bringmann ◽  
Jehanne Dousse ◽  
Jeremy Lovejoy ◽  
Karl Mahlburg
Keyword(s):  
Modular Form ◽  
Generating Function ◽  
Difference Equations ◽  
Circle Method ◽  
Hecke Operators ◽  
Asymptotic Formulas ◽  
Linear Congruences ◽  
The Difference ◽  
Mock Modular Form

We use $q$-difference equations to compute a two-variable $q$-hypergeometric generating function for overpartitions where the difference between two successive parts may be odd only if the larger part is overlined. This generating function specializes in one case to a modular form, and in another to a mixed mock modular form. We also establish a two-variable generating function for the same overpartitions with odd smallest part, and again find modular and mixed mock modular specializations. Applications include linear congruences arising from eigenforms for $3$-adic Hecke operators, as well as asymptotic formulas for the enumeration functions. The latter are proven using Wright's variation of the circle method.

scholarly journals Congruences modulo powers of 5 for the rank parity function

2021 ◽  
Vol Volume 43 - Special... ◽  
Author(s):  
Dandan Chen ◽  
Rong Chen ◽  
Frank Garvan
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Theta Function ◽  
Mock Theta Function ◽  
Parity Function ◽  
International Audience

International audience It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for the rank parity function is f (q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.

scholarly journals Partition Theoretic Interpretation of Two Identities of Euler

2020 ◽  
Vol 19 ◽  
pp. 96-98
Author(s):  
Ananta Kr Bora
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Restricted Partition

In this paper, we have derived a generating function for a restricted partition function. This is in conjunction with two identities of Euler provides a new partition theoretic interpretation of two identities of Euler.

Arithmetic properties of l-regular overpartitions

2016 ◽  
Vol 12 (03) ◽  
pp. 841-852 ◽  
Author(s):  
Erin Y. Y. Shen
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Combinatorial Interpretation ◽  
Arithmetic Properties ◽  
Vector Partitions

Recently, Andrews introduced the partition function [Formula: see text] as the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts [Formula: see text] may be overlined. He proved that [Formula: see text] and [Formula: see text] are divisible by [Formula: see text]. Let [Formula: see text] be the number of overpartitions of [Formula: see text] into parts not divisible by [Formula: see text]. In this paper, we call the overpartitions enumerated by the function [Formula: see text] [Formula: see text]-regular overpartitions. For [Formula: see text] and [Formula: see text], we obtain some explicit results on the generating function dissections. We also derive some congruences for [Formula: see text] modulo [Formula: see text], [Formula: see text] and [Formula: see text] which imply the congruences for [Formula: see text] proved by Andrews. By introducing a rank of vector partitions, we give a combinatorial interpretation of the congruences of Andrews for [Formula: see text] and [Formula: see text].

Crystals and p-adic Integration

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Whittaker Function ◽  
Character Formula ◽  
Whittaker Functions ◽  
Unipotent Subgroup ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebra ◽  
Kostant Partition Function ◽  
Weyl Character Formula

This chapter describes the properties of Kashiwara's crystal and its role in unipotent p-adic integrations related to Whittaker functions. In many cases, integrations of representation theoretic import over the maximal unipotent subgroup of a p-adic group can be replaced by a sum over Kashiwara's crystal. Partly motivated by the crystal description presented in Chapter 2 of this book, this perspective was advocated by Bump and Nakasuji. Later work by McNamara and Kim and Lee extended this philosophy yet further. Indeed, McNamara shows that the computation of the metaplectic Whittaker function is initially given as a sum over Kashiwara's crystal. The chapter considers Kostant's generating function, the character of the quantized enveloping algebra, and its association with Kashiwara's crystal, along with the Kostant partition function and the Weyl character formula.

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