Jesús Guillera
We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $π$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many primes. In addition we are able to prove a few of them from some terminating hypergeometric identities. Finally, we make an intriguing observation.
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We define bilateral series related to Ramanujan-like series for $1/π^2$. Then, we conjecture a property of them and give some applications.
Using results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsom, Ono, and Rhoades for mock theta functions. Here we see that the author's previous work on the dual nature of Appell--Lerch sums and partial theta functions and on constructing bilateral $q$-series with mixed mock modular behaviour is well-suited for such radial limits. We present five more radial limit results which follow from mixed mock modular bilateral $q$-hypergeometric series. We also obtain the mixed mock modular bilateral series for a universal mock theta function…
We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for explicit examples of Siegel modular forms.
In contrast to all other known Ramanujan-type congruences, we discover that Ramanujan-type congruences for Hurwitz class numbers can be supported on non-holomorphic generating series. We establish a divisibility result for such non-holomorphic congruences of Hurwitz class numbers. The two keys tools in our proof are the holomorphic projection of products of theta series with a Hurwitz class number generating series and a theorem by Serre, which allows us to rule out certain congruences.
Recently Lachterman, Schayer, and Younger published an elegant proof of the Ramanujan congruences for the partition function $p(n)$. Their proof uses only the classical theory of modular forms as well as a beautiful result of Choie, Kohnen, and Ono, without need for Hecke operators. In this paper we give a method for generalizing Lachterman, Schayer, and Younger's proof to include Ramanujan congruences for multipartition functions $p_k(n)$, and Ramanujan congruences for $p(n)$ modulo certain prime powers.
We prove several Ramanujan-type congruences modulo powers of $5$ for partition $k$-tuples with $5$-cores, for $k=2, 3, 4$. We also prove some new infinite families of congruences modulo powers of primes for $k$-tuples with $p$-cores, where $p$ is a prime.
In this paper, we prove supercongruence relations for truncated $N$-tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogs of some Ramanujan-type supercongruences.
For any non-negative integer $n$ and non-zero integer $r$, let $p_r(n)$ denote Ramanujan's general partition function. By employing $q$-identities, we prove some new Ramanujan-type congruences modulo 5 for $p_r(n)$ for $r=-(5λ+1), -(5λ+3), -(5λ+4), -(25λ+1)$, $-(25λ+2)$, and any integer $λ$.
We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific congruences for Hecke eigenvalues. In particular, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra. More generally, we show for weakly holomorphic modular forms of integral weight, that Ramanujan-type congruences naturally occur for shifts in the union of two square-classes as opposed to single square-classes that appear in the literature on the partition function. We also rule out the possibility of s…
For the unrestricted partition function $p(n)$ for integers $n \geq 0$, it is known that $p(49n + 19) \equiv 0 \pmod{49}$, $p(49n + 33) \equiv 0 \pmod{49}$, and $p(49n + 40) \equiv 0 \pmod{49}$ for all $n \geq 0$. We find witness identities for these Ramanujan congruences.
Ramanujan's last letter to Hardy explored the asymptotic properties of modular forms, as well as those of certain interesting $q$-series which he called \emph{mock theta functions}. For his mock theta function $f(q)$, he claimed that as $q$ approaches an even order $2k$ root of unity $ζ$, \[\lim_{q\to ζ} \big(f(q) - (-1)^k (1-q)(1-q^3)(1-q^5)\cdots (1-2q + 2q^4 - \cdots)\big) = O(1),\] and hinted at the existence of similar statements for his other mock theta functions. Recent work of Folsom-Ono-Rhoades provides a closed formula for the implied constant in this radial limit of $f(q)$. Here, by…
New Ramanujan-type congruences for overpartitions modulo $11$ and $13$
本文通过模形式理论推导出11和13模下的新分拆恒等式，并推测其他模数的可能恒等式。
In the case of Siegel modular forms of degree $n$, we prove that, for almost all prime ideals $\frak{p}$ in any ring of algebraic integers, mod $\frak{p}^m$ cusp forms are congruent to true cusp forms of the same weight. As an application of this property, we give congruences for the Klingen-Eisenstein series and cusp forms, which can be regarded as a generalization of Ramanujan's congruence. We will conclude by giving numerical examples.
Ramanujan famously found congruences for the partition function like p(5n+4) = 0 modulo 5. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Gamma_{1}(4) which 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.
In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, $A_1(m)$, which counted subpartitions with a structure related to the Rogers--Ramanujan identities. They conjectured the existence of an infinite class of congruences for $A_1(m)$, modulo powers of 5. We give an explicit form of this conjecture, and prove it for all powers of 5.
We show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra and provide several structure results for them. We discover a dichotomy between congruences originating in Hecke eigenvalues and congruences on arithmetic progressions with cube-free periods. The scarcity of the latter was investigated recently. We explain that they provide congruences among algebraic parts of twisted central $\mathrm{L}$-values. We specialize our results to partition congruences, for which we investigate the proofs of partition congruences by Atkin and by Ono, and develop a heuris…
Let $B_{l,m}(n)$ denote the number of $(l,m)$-regular bipartitions of $n$. Recently, many authors proved several infinite families of congruences modulo $3$, $5$ and $11$ for $B_{l,m}(n)$. In this paper, using theta function identities to prove infinite families of congruences modulo $m$ for $(l,m)$-regular bipartitions, where $m\in\{7,3,11,13,17\}$.
We show that if $E/\mathbb{Q}$ is an elliptic curve with a rational $p$-torsion for $p=2$ or $3$, then there is a congruence relation between Ramanujan's tau function and $E$ modulo $p$. We make use of such congruences to compute the Iwasawa invariants of $2$-adic and $3$-adic Mazur--Tate elements attached to Ramanujan's tau function. We also investigate numerically the Iwasawa invariants of the Mazur--Tate elements attached to an elliptic curve with additive reduction at a fixed prime number.
Recently, Garthwaite-Penniston have shown that the coefficients of Ramanujan's mock theta function $ω$ satisfy infinitely many congruences of Ramanujan-type. In this work we give the first explicit examples of congruences for Ramanujan's mock theta function $ω$ and another mock theta function $\mathcal{C}$.
Universal Ramanujan-type congruences for prime-detecting quasimodular forms
该研究证明了素数检测拟模形式全Z-模上存在无穷多个通用拉马努金型同余式，且对每个素数p均存在对应算术级数使相关系数被p整除。
Some New Congruences and Partition-Theoretic Interpretations for the Coefficients of Some Rogers-Ramanujan Type Identities
本文利用正整数的 overpartition 和 colour partition，为某些 Rogers-Ramanujan 型恒等式系数提供了分拆理论解释，并证明了模 2 的幂的无限族同余。
Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical results on congruences of modular forms to obtain congruences for Maass forms. In this note we work out the example of Ramanujan's mock theta functions f and ω in detail.
We prove congruences between cuspidal newforms and Eisenstein series of prime level, which generalize Ramanujan's congruence. Such congruences were recently found by Billerey and Menares, and we refine them by specifying the Atkin-Lehner eigenvalue of the newform involved. We show that similar refinements hold for the level raising congruences between cuspidal newforms of different levels, due to Ribet and Diamond. The proof relies on studying the new subspace and the Eisenstein subspace of the space of period polynomials for the congruence subgroup $Γ_0(N)$, and on a version of Ihara's lemma.
Bilateral identities of the Rogers--Ramanujan type
本文利用解析方法建立一系列 Rogers–Ramanujan 型双边及多边恒等式，推广多项经典恒等式，并给出闭式求和。
Explainable Ramanujan-type Congruences on Square-Classes of Arithmetic Progressions
本文利用带挠点奇点的 Jacobi 形式及模群表示，形式化“可解释 Ramanujan 型同余”，并证明其出现在等差数列平方类上。
We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence of a prime dividing the $k-1$-th generalized Bernoulli number and the existence of non-trivial Hermitian cusp forms of weight $k$. We will conclude by giving numerical examples for each case.
Cranks for Ramanujan-type congruences of $k$-colored partitions
本文为分拆函数同余族的不变量提供研究框架，利用θ块理论找到一族曲柄函数，可解释多数已知k着色分拆函数同余，具理论价值。
We establish Ramanujan-style congruences modulo certain primes $\ell$ between an Eisenstein series of weight $k$, prime level $p$ and a cuspidal newform in the $\varepsilon$-eigenspace of the Atkin-Lehner operator inside the space of cusp forms of weight $k$ for $Γ_0(p)$. Under a mild assumption, this refines a result of Gaba-Popa. We use these congruences and recent work of Ciolan, Languasco and the third author on Euler-Kronecker constants, to quantify the non-divisibility of the Fourier coefficients involved by $\ell.$ The degree of the number field generated by these coefficients we invest…
We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us to establish Ramanujan-type congruences for Hurwitz class numbers on square classes, where the holomorphic case parallels previous work by Radu on partition congruences. We offer two applications. The first application demonstrates common divisibility features of Ramanujan-type congruences for Hurwitz class numbers. The second application provides a dichotomy between congruences for class numbers of imaginary quadra…
On the occurrence of congruence multiplicities between Ramanujan's theta functions
本研究证明了拉马努金θ函数φ(-q)与ψ(q)相关无穷级数的八类模3、5任意次幂的同余关系，明确二者同余族的同构性并预测额外同构的存在。
We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes $\ell$ for which their coefficients $c(n)$ obey congruences of the form $c(\ell n + a) \equiv 0 \pmod \ell$. We apply this result to obtain a complete characterization of the congruences of the same form that the sequences $c_N(n)$ satisfy, where $c_N(n)$ is defined by $ \sum_{n=0}^{\infty} c_N(n)q^n = \prod_{n=1}^{\infty} \frac{1}{(1-q^n)(1-q^{Nn})}$. This last result answers a question of H.-C. Chan.
Consider the linear congruence equation $x_1+\ldots+x_k \equiv b\,(\text{mod } n)$ for $b,n\in\mathbb{Z}$. By $(a,b)_s$, we mean the largest $l^s\in\mathbb{N}$ which divides $a$ and $b$ simultaneously. For each $d_j|n$, define $\mathcal{C}_{j,s} = \{1\leq x\leq n^s | (x,n^s)_s = d^s_j\}$. Bibak et al. gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on $x_i$. We generalize their result with generalized gcd restrictions on $x_i$ by proving that for the above linear congruence, the number of solutions is $$\frac{1}{n^s}\s…
The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we investigate the existence and classification of Ramanujan-type congruences for functions in multiplicative number theory.
Heuristic derivation of Zudilin's supercongruences for rational Ramanujan series
本文用启发式方法推导出双边拉马努金级数的$p$-进伴侣，并由此得到Zudilin超同余式。
We investigate Ramanujan congruences for the function which counts the overpartitions of n with restricted odd differences. In particular, we show that only one such congruence exists. Our method involves using the theory of modular forms to prove a more general theorem which bounds the number of primes possible for Ramanujan congruences in certain eta-quotients. This generalizes work done by Jonah Sinick. We also provide two congruences modulo 5 for this function.
Let ${p}_{3,3}(n)$ denote the number of $2$-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $3$. In this paper, we shall establish some interesting Ramanujan-type congruences for ${p}_{3,3}(n)$.
In this paper we obtain asymptotic formulas for the Fourier coefficients of an infinite family of crank generating functions. Moreover we use this result to show that the crank obeys certain inequalities. This implies that the crank can not explain any partition congruences in the usual way beside the three deduced by Ramanujan.
Lambda-invariants of Mazur--Tate elements attached to Ramanujan's tau function and congruences with Eisenstein series
该研究针对权12模形式$\Delta$，证明其与特定艾森斯坦级数模$p$（$p\in\{3,5,7\}$）同余，并推导了相关Mazur–Tate元的岩泽不变量公式，验证了此前的数值数据。
Congruences are found modulo powers of 5, 7 and 13 for Andrews' smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan's partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit Ramanujan-type congruences for spt(n) modulo p for all primes p>3 which were conjectured earlier by the author. We extend Ono's method to handle the powers of 5, 7 and 13 congruences. We need the theory of weak Maass forms as well as certain classical modular equations for the Dedekind eta-function.
Explicit constructions for Ramanujan-type congruences
本文通过模形式的显式构造，为包括eta商、弱全纯模形式和mock模形式在内的生成函数建立Ramanujan型同余式，统一恢复已知同余式并得到新的非同余结果与显式同余式。
We considerably improve Ono's and Ahlgren-Ono's work on the frequent occurrence of Ramanujan-type congruences for the partition function, and demonstrate that Ramanujan-type congruences occur in families that are governed by square-classes. We thus elucidate for the first time an exemplary family of congruences found by Atkin-O'Brien. Our results are based on a novel framework that leverages available results on integral models of modular curves via representations of finite quotients of $\mathrm{SL}_2(\mathbb{Z})$ or $\mathrm{Mp}_1(\mathbb{Z})$. This framework applies to congruences of all we…
This paper aims to introduce two systems of nonlinear ordinary differential equations whose solution components generate the graded algebra of quasi-modular forms on Hecke congruence subgroups $Γ_0(2)$ and $Γ_0(3)$. Using these systems, we provide the generated graded algebras with an $\mathfrak{sl}_2(\mathbb{C})$-module structure. As applications, we introduce Ramanujan-type tau functions for $Γ_0(2)$ and $Γ_0(3)$, and obtain some interesting and non-trivial recurrence and congruence relations.
For rational $α$, the fractional partition functions $p_α(n)$ are given by the coefficients of the generating function $(q;q)^α_\infty$. When $α=-1$, one obtains the usual partition function. Congruences of the form $p(\ell n + c)\equiv 0 \pmod{\ell}$ for a prime $\ell$ and integer $c$ were studied by Ramanujan. Such congruences exist only for $\ell\in\{5,7,11\}.$ Chan and Wang [4] recently studied congruences for the fractional partition functions and gave several infinite families of congruences using identities of the Dedekind eta-function. Following their work, we use the theory of non-ord…
Let $\ell$ be any fixed prime number. We define the $\ell$-Genocchi numbers by $G_n:=\ell(1-\ell^n)B_n$, with $B_n$ the $n$-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes. We say that an odd prime $p$ is $\ell$-Genocchi irregular if it divides at least one of the $\ell$-Genocchi numbers $G_2,G_4,\ldots, G_{p-3}$, and $\ell$-regular otherwise. With the help of techniques used in the study of Artin's primitive root conjecture, we give asymptotic estimates for the number of $\ell$-Genocchi irregular primes in a prescribed arithm…
Ramanujan's congruence primes
本研究受拉马努金关于尖点形式傅里叶系数与素数691的同余结论启发，发现素数67出现在模5二次狄利克雷特征的L函数相关比值的分子中，导出了$Γ_0(5)$上权6尖点形式与艾森斯坦级数的模67同余关系。
Ramanujan type congruences for quotients of Klein forms
本文基于Klein形式商构造level p模形式乘积，导出了素数p≥5幂次的Ramanujan型同余，完整列举了特定权与级数下的同余乘积，并构建了模形式分次代数的生成集与级数分解方法。
An easy proof of Ramanujan's famous congruences $p(5m+4)\equiv 0 \equiv τ(5m+5) \pmod 5$
本文利用 Jacobi 三重积恒等式的极限情形和拉马努金计算 $τ(n)$ 的技巧，给出其两个模 5 同余式的简明证明。
Sign Patterns and Congruences of certain infinite products involving the Rogers-Ramanujan continued fraction
本文研究涉及Rogers-Ramanujan连分数的无穷乘积系数符号规律，证明特定算术级数中系数的正负性，并发现若干模3、模4和模15的同余式。
When Mike Hirschhorn showed us his lovely gem, that gives the simplest-to-date proof of Ramanujan's famous result that p(11n+6) is divisible by 11, we realized that his amazing method can be extended, and taught to a computer, and can prove even deeper identities. We would have done much more if not for the existence of Silviu Radu's powerful algorithm that handle any Ramanujan type congruence for any modular form (of a very general type), but it is still nice to know that in order to prove such simply stated results, that can be explained to a seven-year-old, one does not need the intimidatin…
Let $b_\ell(n)$ be the number of $\ell$-regular partitions of $n$. Recently, Hou et al established several infinite families of congruences for $b_\ell(n)$ modulo $m$, where $(\ell,m)=(3,3),(6,3),(5,5),(10,5)$ and $(7,7)$. In this paper, by the vanishing property given by Hou et al, we show an infinite family of congruence for $b_{11}(n)$ modulo $11$. Moreover, for $\ell= 3, 13$ and $25$, we obtain three infinite families of congruences for $b_{\ell}(n)$ modulo $3, 5$ and $13$ by the theory of Hecke eigenforms.
Arithmetic properties of generalized Frobenius partitions
本文研究了广义m色Frobenius分拆函数cϕ_m在m为5、7、11时的同余性质，结合模形式理论与计算证明了仅有六个类似Ramanujan的同余式存在。
This article considers the eta power $\prod {(1-q^k)}^{b-1}$. It is proved that the coefficients of $\frac{q^n}{n!}$ in this expression, as polynomials in $b$, exhibit equidistribution of the coefficients in the nonzero residue classes mod 5 when $n=5j+4$. Other symmetries, as well as symmetries for other primes and prime powers, are proved, and some open questions are raised.
Congruences for Andrews' Smallest Parts Partition Function and New Congruences for Dyson's Rank
设spt（n）表示n的划分中最小部分出现的总数。最近，Andrews证明了spt（n）与二阶矩的关系，并证明了一些令人惊讶的Ramanujan型模5，7和13同余。我们证明了这些同余式的推广使用已知的秩和曲柄矩之间的关系。我们得到了spt（n）mod p（p = 11，17，19，29，31和37）的显式Ramanujan型同余.最近，Bringmann和Ono证明了Dyson秩函数有无穷多个Ramanujan型同余。他们的证明是非构造性的，并利用弱Maass形式理论。我们构造两个显式的非平凡的例子模11使用之间的秩矩和半整数权重Hecke特征形的初等同余。
The notion of cubic partitions is introduced by Hei-Chi Chan and named by Byungchan Kim in connection with Ramanujan's cubic continued fractions. Chan proved that cubic partition function has Ramanujan Type congruences modulo powers of $3$. In a recent paper, William Y.C. Chen and Bernard L.S. Lin studied the congruent property of the cubic partition function modulo $5$. In this note, we give Ramanujan type congruences for cubic partition function modulo powers of $5$.
Ramanujan's theta functions and internal congruences modulo arbitrary powers of $3$
本文研究了源自Ramanujan经典theta函数的两个函数模$3$任意幂的内部同余，证明了具体的同余关系式，大幅推广了前人的相关结果。
We introduce two new integer partition functions, both of which are the number of partition quadruples of $n$ with certain size restrictions. We prove both functions satisfy Ramanujan-type congruences modulo $3$, $5$, $7$, and $13$ by use of generalized Lambert series identities and $q$-series techniques.
For several congruence subgroups of low levels and their conjugates, we derive differential equations satisfied by the Eisenstein series of weight 4 and relate them to elliptic curves, whose associated new forms of weight 2 constitute the list of Martin and Ono of new forms given by eta-products/quotients.
The cubic partitions of a natural number $n$, introduced by Chan and Kim, have generating function $\sum_{n=0}^{\infty}a(n)q^n= \frac{1}{(q; q)_{\infty}(q^2; q^2)_{\infty}}.$ In this paper, we generalize some results of Chen-Lin, which suggest that $a(n)$ should have analogous properties of the ordinary partition function. Specifically, we show that for every non-negative integer $n$, $a(5^4n+547)\equiv 0\pmod{5^2}, a(7^3n+190)\equiv 0\pmod{7^2}, a(7^3n+288 \equiv 0\pmod{7^2} and a(7^3n+337)\equiv 0\pmod{7^2}.$
Linear Congruences in several variables with congruence restrictions
本文研究带限制条件的多元线性同余方程组，利用 Ramanujan 和给出解数的充要条件与显式公式，并推广至 $\mathbb{F}_q[t]$ 函数域。
Proofs of two $q$-congruence conjectures of Guo
本文证明了Guo提出的两个q-同余猜想，分别涉及截断q-模拟的平方分圆同余和带参数s的截断基本超几何和的可除性。
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