Ramanujan showed that $τ(p) \equiv p^{11}+1 \pmod{691}$, where $τ(n)$ is the $n$-th Fourier coefficient of the unique normalized cusp form of weight $12$ and full level, and the prime $691$ appears in the numerator of $ζ(12)/π^{12}$ for the Riemann zeta function $ζ(s)$. Searching for such congruences, it is shown that the prime $67$ appears in the numerator of $L(6,χ)/(π^6 \sqrt{5})$, where $χ$ is the unique nontrivial quadratic Dirichlet character modulo $5$ and $L(s,χ)$ its Dirichlet $L$-function, giving rise to a congruence $f_χ\equiv E^\circ_{6, χ} \pmod{67}$ between a cusp form $f_χ$ and an Eisenstein series $E^\circ_{6, χ}$ of weight $6$ on $Γ_0(5)$ with nebentypus character $χ.$
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.
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 $\mathcal{C}$.
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 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.
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.
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.
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…
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.
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.
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…
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.
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.
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…
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.
We study the Ramanujan-prime-counting function along the lines of Ramanujan's original work on Bertrand's Postulate. We show that the number of Ramanujan primes between x and 2x tends to infinity with x. This analysis leads us to define a new sequence of prime numbers, which we call derived Ramanujan primes. For n greater than or equal to 1 we define the nth derived Ramanujan prime as the smallest positive integer with the property that if x is greater than or equal to the nth derived Ramanujan prime, then the number of Ramanujan primes in the interval from x to x/2 is greater than or equal to…
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…
Considering Ramanujan primes and the symmetric to them so-called Labos primes, we study their parallel properties, we study all primes with these properties (generalized Ramanujan and Labos primes) and construct two kinds of sieves for them. Finally, we give a further natural generalization of these constructions and pose some conjectures and open problems.
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.
In this paper we establish an explicit upper bound for the first $k$-Ramanujan prime $R_1^{(k)}$ by using a recent result concerning the existence of prime numbers in small intervals.
In this paper we establish several results concerning the generalized Ramanujan primes. For $n\in\mathbb{N}$ and $k \in \mathbb{R}_{> 1}$ we give estimates for the $n$th $k$-Ramanujan prime which lead both to generalizations and to improvements of the results presently in the literature. Moreover, we obtain results about the distribution of $k$-Ramanujan primes. In addition, we find explicit formulae for certain $n$th $k$-Ramanujan primes. As an application, we prove that a conjecture of Mitra, Paul and Sarkar concerning the number of primes in certain intervals holds for every sufficiently la…
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.
The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x \geq R_n$ the interval $(x/2, x]$ contains at least $n$ primes. In this paper we undertake a study of the sequence $(π(R_n))_{n \in \mathbb{N}}$, which tells us where the $n$th Ramanujan prime appears in the sequence of all primes. In the first part we establish new explicit upper and lower bounds for the number of primes up to the $n$th Ramanujan prime, which imply an asymptotic formula for $π(R_n)$ conjectured by Yang and Togbé. In the second part of this paper, we use these explicit estimates to derive a r…
In this paper, we give a new upper bound for the number $N_{\mathcal{R}}$ which is defined to be the smallest positive integer such that a certain inequality due to Ramanujan involving the prime counting function $π(x)$ holds for every $x \geq N_{\mathcal{R}}$.
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 p for p = 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 theor…
A prime p is called Sophie Germain prime if 2p+1 is also prime. A formula for the density of such primes is given in a more general setting using a new approach. This method uses the Ramanujan-Fourier series for a modified von Mangoldt function. The proof remains heuristic as interchange of certain limits has not been justified. Experimental evidence using computer calculations is provided for the plausibility of the result.
The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then the interval $(x/2,x]$ contains at least $n$ primes. We sharpen Laishram's theorem that $R_n < p_{3n}$ by proving that the maximum of $R_n/p_{3n}$ is $R_5/p_{15} = 41/47$. We give statistics on the length of the longest run of Ramanujan primes among all primes $p<10^n$, for $n\le9$. We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below $10^n$ of three types. Finally, we relate runs of Ramanujan primes to prime gaps. Along the way we st…
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 $5$.
In this paper, we propose the use of Ramanujan class of polynomials for the construction of prime order elliptic curves using the CM-method. We compare (theoretically and experimentally) the efficiency of using this new class against the use of the Weber, $M_{D,l}(x)$ and $M_{D,p_1,p_2}(x)$ polynomials and show that they clearly outweigh all of them in the generation of prime order elliptic curves.
In this paper we use refined approximations for Chebyshev's $\vartheta$-function to establish new explicit estimates for the prime counting function $π(x)$, which improve the current best estimates for large values of $x$. As an application we find an upper bound for the number $H_0$ which is defined to be the smallest positive integer so that Ramanujan's prime counting inequality holds for every $x \geq H_0$.
