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An Analysis Of Antichimeral Ramanujan Type Congruences For Quotients Of The Rogers-Ramanujan Functions, Ryan A. Mowers
An Analysis Of Antichimeral Ramanujan Type Congruences For Quotients Of The Rogers-Ramanujan Functions, Ryan A. Mowers
Theses and Dissertations
This paper proves the existence of antichimeral Ramanujan type congruences for certain modular forms These modular forms can be represented in terms of Klein forms and the Dedekind eta function. The main focus of this thesis is to introduce the necessary theory to characterize these specific Ramanujan type congruences and prove their antichimerality.
Congruences For Quotients Of Rogers-Ramanujan Functions, Maria Del Rosario Valencia Arevalo
Congruences For Quotients Of Rogers-Ramanujan Functions, Maria Del Rosario Valencia Arevalo
Theses and Dissertations
In 1919 the mathematician Srinivasa Ramanujan conjectured congruences for the partition function p(n) modulo powers of the primes 5,7,11. In this work, we study Ramanujan type congruences modulo powers of primes p = 7,11,13,17,19,23 satisfied by the Fourier coefficients of quotients the Rogers-Ramanujan Functions G(τ) and H(τ) and the Dedekind eta function η(5τ). In addition to deriving new congruences, we develop the foundational theory of modular forms to motivate and prove the results. The work includes proofs of congruences facilitated by Python/SageMath code.
Cranks For Partitions With Bounded Largest Part, Dennis Eichhorn, Brandt Kronholm, Acadia Larsen
Cranks For Partitions With Bounded Largest Part, Dennis Eichhorn, Brandt Kronholm, Acadia Larsen
School of Mathematical and Statistical Sciences Faculty Publications and Presentations
For 80 years, Dyson’s rank has been known as the partition statistic that witnesses the first two of Ramanujan’s celebrated congruences for the ordinary partition function. In this paper, we show that Dyson’s rank actually witnesses families of partition congruences modulo every prime . This comes from an in-depth study of when a “multiplicity-based statistic” is a crank witnessing congruences for the function p ` n, m˘ , which enumerates partitions of n with parts of size at most m. We also show that as the modulus increases, there is an ever-growing collection of distinct multiplicity-based cranks witnessing these same …
Arithmetic Properties Of Septic Partition Functions, Timothy Huber, Mayra Huerta, Nathaniel Mayes
Arithmetic Properties Of Septic Partition Functions, Timothy Huber, Mayra Huerta, Nathaniel Mayes
School of Mathematical and Statistical Sciences Faculty Publications and Presentations
Congruences and related identities are derived for a set of colored and weighted partition functions whose generating functions generate the graded algebra of integer weight modular forms of level seven. The work determines a general strategy for identifying and proving identities and associated congruences for modular forms on the principal congruence subgroup of level 7. Ramanujan's partition congruence modulo 7 serves as a prototype for the process used to prove new congruences for modular forms of level 7.
On Congruences Related To Trinomial Coefficients And Harmonic Numbers, Neşe Ömür, Si̇bel Koparal, Laid Elkhiri
On Congruences Related To Trinomial Coefficients And Harmonic Numbers, Neşe Ömür, Si̇bel Koparal, Laid Elkhiri
Turkish Journal of Mathematics
In this paper, we establish some congruences involving the trinomial coefficients and harmonic numbers. For example, for any prime $p>3,$ \begin{equation*} \sum\limits_{k=0}^{p-1}\left( -1\right) ^{k}\binom{p-1}{k}_{2}H_{k}\equiv 0 \ \pmod {p}. \end{equation*}
Congruences For Coefficients Of Modular Functions In Levels 3, 5, And 7 With Poles At 0, Ryan Austin Keck
Congruences For Coefficients Of Modular Functions In Levels 3, 5, And 7 With Poles At 0, Ryan Austin Keck
Theses and Dissertations
We give congruences modulo powers of p in {3, 5, 7} for the Fourier coefficients of certain modular functions in level p with poles only at 0, answering a question posed by Andersen and Jenkins and continuing work done by the Jenkins, the author, and Moss. The congruences involve a modulus that depends on the base p expansion of the modular form's order of vanishing at infinity.
Congruences For Fourier Coefficients Of Modular Functions Of Levels 2 And 4, Eric Brandon Moss
Congruences For Fourier Coefficients Of Modular Functions Of Levels 2 And 4, Eric Brandon Moss
Theses and Dissertations
We give congruences modulo powers of 2 for the Fourier coefficients of certain level 2 modular functions with poles only at 0, answering a question posed by Andersen and Jenkins. The congruences involve a modulus that depends on the binary expansion of the modular form's order of vanishing at infinity. We also demonstrate congruences for Fourier coefficients of some level 4 modular functions.
On The Density Of The Odd Values Of The Partition Function, Samuel Judge
On The Density Of The Odd Values Of The Partition Function, Samuel Judge
Dissertations, Master's Theses and Master's Reports
The purpose of this dissertation is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo $2$. We provide a doubly-indexed, infinite family of conjectural identities in the ring of series $\Z_2[[q]]$, which relate $p(n)$ with suitable $t$-multipartition functions, and show how to, in principle, prove each such identity. We will exhibit explicit proofs for $32$ of our identities. However, the conjecture remains open in full generality. A striking consequence of these conjectural identities is that, under suitable …
Arithmetic Properties Of $\Ell$-Regular Overpartition Pairs, Megadahalli Siddanaika Mahadeva Naika, Chandrappa Shivashankar
Arithmetic Properties Of $\Ell$-Regular Overpartition Pairs, Megadahalli Siddanaika Mahadeva Naika, Chandrappa Shivashankar
Turkish Journal of Mathematics
In this paper, we investigate the arithmetic properties of $\ell$-regular overpartition pairs. Let $\overline{B}_{\ell}(n)$ denote the number of $\ell$-regular overpartition pairs of $n$. We will prove the number of Ramanujan-like congruences and infinite families of congruences modulo 3, 8, 16, 36, 48, 96 for $\overline{B}_3(n)$ and modulo 3, 16, 64, 96 for $\overline{B}_4(n)$. For example, we find that for all nonnegative integers $\alpha$ and $n$, $\overline{B}_{3}(3^{\alpha}(3n+2))\equiv 0\pmod{3}$, $\overline{B}_{3}(3^{\alpha}(6n+4))\equiv 0\pmod{3}$, and $\overline{B}_{4}(8n+7)\equiv 0\pmod{64}$.
Weakly Holomorphic Modular Forms In Prime Power Levels Of Genus Zero, David Joshua Thornton
Weakly Holomorphic Modular Forms In Prime Power Levels Of Genus Zero, David Joshua Thornton
Theses and Dissertations
Let N ∈ {8,9,16,25} and let M#0(N) be the space of level N weakly holomorphic modular functions with poles only at the cusp at infinity. We explicitly construct a canonical basis for M#0(N) indexed by the order of the pole at infinity and show that many of the coefficients of the elements of these bases are divisible by high powers of the prime dividing the level N. Additionally, we show that these basis elements satisfy an interesting duality property. We also give an argument that extends level 1 results …
Obstruction Criteria For Modular Deformation Problems, Jeffrey Hatley Jr
Obstruction Criteria For Modular Deformation Problems, Jeffrey Hatley Jr
Doctoral Dissertations
For a cuspidal newform f of weight k at least 3 and a prime p of the associated number field Kf, the deformation problem for its associated mod p Galois representation is unobstructed for all primes outside some finite set. Previous results gave an explicit bound on this finite set for f of squarefree level; we modify this bound and remove the squarefree hypothesis. We also show that if the p-adic deformation problem for f is unobstructed, then f is not congruent mod p to a newform of lower level.