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.

POWERS OF THE ETA-FUNCTION AND HECKE OPERATORS

2012 ◽  
Vol 08 (03) ◽  
pp. 599-611 ◽  
Author(s):  
ALEXANDER CARNEY ◽  
ANASTASSIA ETROPOLSKI ◽  
SARAH PITMAN
Keyword(s):  
Partition Function ◽  
Large Class ◽  
Delta Function ◽  
Hecke Operators ◽  
Faber Polynomials ◽  
Partition Functions ◽  
Fast Calculation ◽  
Half Integer ◽  
Integer Weight

Half-integer weight Hecke operators and their distinct properties play a major role in the theory surrounding partition numbers and Dedekind's eta-function. Generalizing the work of Ono in [K. Ono, The partition function and Hecke operators, Adv. Math.228 (2011) 527–534], here we obtain closed formulas for the Hecke images of all negative powers of the eta-function. These formulas are generated through the use of Faber polynomials. In addition, congruences for a large class of powers of Ramanujan's Delta-function are obtained in a corollary. We further exhibit a fast calculation for many large values of vector partition functions.

SYMMETRY INSIGHTS FOR DESIGN OF SUPERCOMPUTER NETWORK TOPOLOGIES: ROOTS AND WEIGHTS LATTICES

2012 ◽  
Vol 26 (31) ◽  
pp. 1250169 ◽  
Author(s):  
YUEFAN DENG ◽  
ALEXANDRE F. RAMOS ◽  
JOSÉ EDUARDO M. HORNOS
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Root Systems ◽  
Hardware Architecture ◽  
Classification Theorem ◽  
Simple Complex ◽  
Half Integer ◽  
Network Topologies ◽  
Integer Weight

We present a family of networks whose local interconnection topologies are generated by the root vectors of a semi-simple complex Lie algebra. Cartan classification theorem of those algebras ensures those families of interconnection topologies to be exhaustive. The global arrangement of the network is defined in terms of integer or half-integer weight lattices. The mesh or torus topologies that network millions of processing cores, such as those in the IBM BlueGene series, are the simplest member of that category. The symmetries of the root systems of an algebra, manifested by their Weyl group, lends great convenience for the design and analysis of hardware architecture, algorithms and programs.

On Hecke eigenvalues of Siegel modular forms in the Maass space

2018 ◽  
Vol 30 (3) ◽  
pp. 775-783 ◽  
Author(s):  
Sanoli Gun ◽  
Biplab Paul ◽  
Jyoti Sengupta
Keyword(s):  
Modular Forms ◽  
Absolute Constant ◽  
Modular Group ◽  
Siegel Modular Forms ◽  
Maass Forms ◽  
Formula One ◽  
Hecke Eigenvalues ◽  
Hecke Eigenforms ◽  
The Third ◽  
Limit Points

AbstractIn this article, we prove an Omega result for the Hecke eigenvalues {\lambda_{F}(n)} of Maass forms F which are Hecke eigenforms in the space of Siegel modular forms of weight k, genus two for the Siegel modular group {Sp_{2}({\mathbb{Z}})}. In particular, we prove\lambda_{F}(n)=\Omega\biggl{(}n^{k-1}\exp\biggl{(}c\frac{\sqrt{\log n}}{\log% \log n}\biggr{)}\biggr{)},when {c>0} is an absolute constant. This improves the earlier result\lambda_{F}(n)=\Omega\biggl{(}n^{k-1}\biggl{(}\frac{\sqrt{\log n}}{\log\log n}% \biggr{)}\biggr{)}of Das and the third author. We also show that for any {n\geq 3}, one has\lambda_{F}(n)\leq n^{k-1}\exp\biggl{(}c_{1}\sqrt{\frac{\log n}{\log\log n}}% \biggr{)},where {c_{1}>0} is an absolute constant. This improves an earlier result of Pitale and Schmidt. Further, we investigate the limit points of the sequence {\{\lambda_{F}(n)/n^{k-1}\}_{n\in{\mathbb{N}}}} and show that it has infinitely many limit points. Finally, we show that {\lambda_{F}(n)>0} for all n, a result proved earlier by Breulmann by a different technique.

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.

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