Ramanujan-style congruences of local origin

International Journal of Number Theory

We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ℓ for which their coefficients c(n) obey congruences of the form c(ℓn + a) ≡ 0 ( mod ℓ). We apply this result to obtain a complete characterization of the congruences of the same form that the sequences cN(n) satisfy, where cN(n) is defined by [Formula: see text]. This last result answers a question of H.-C. Chan.

International Mathematics Research Notices, 2010

Mathematische Zeitschrift

We establish Ramanujan-style congruences modulo certain primes ℓ between an Eisenstein series of weight k, prime level p and a cuspidal newform in the ε-eigenspace of the Atkin-Lehner operator inside the space of cusp forms of weight k for Γ0(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 ℓ. The degree of the number field generated by these coefficients we investigate using recent results on prime factors of shifted prime numbers.

Ramanujan Journal, 2009

In this paper we present an algorithm that takes as input a generating function of the form \(\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n}\) and three positive integers m,t,p, and which returns true if \(a(mn+t)\equiv0\pmod{p},n\geq0\) , or false otherwise. Our method builds on work by Rademacher (Trans. Am. Math. Soc. 51(3):609–636, 1942), Kolberg (Math. Scand. 5:77–92, 1957), Sturm (Lecture Notes in Mathematics, pp. 275–280, Springer, Berlin/Heidelberg, 1987), Eichhorn and Ono (Proceedings for a Conference in Honor of Heini Halberstam, pp. 309–321, 1996).

Proceedings of the National Academy of Sciences, 2005

Ramanujan type congruences.

Journal of Number Theory, 2015

Let p(n) denote the number of overpartitions of n. Recently, Fortin-Jacob-Mathieu and Hirschhorn-Sellers independently obtained 2-, 3-and 4-dissections of the generating function for p(n) and derived a number of congruences for p(n) modulo 4, 8 and 64 including p(5n + 2) ≡ 0 (mod 4), p(4n + 3) ≡ 0 (mod 8) and p(8n + 7) ≡ 0 (mod 64). By employing dissection techniques, Yao and Xia obtained congruences for p(n) modulo 8, 16 and 32, such as p(48n + 26) ≡ 0 (mod 8), p(24n + 17) ≡ 0 (mod 16) and p(72n+69) ≡ 0 (mod 32). In this paper, we give a 16-dissection of the generating function for p(n) modulo 16 and we show that p(16n + 14) ≡ 0 (mod 16) for n ≥ 0. Moreover, by using the 2-adic expansion of the generating function of p(n) due to Mahlburg, we obtain that p(ℓ 2 n + rℓ) ≡ 0 (mod 16), where n ≥ 0, ℓ ≡ −1 (mod 8) is an odd prime and r is a positive integer with ℓ ∤ r. In particular, for ℓ = 7, we get p(49n + 7) ≡ 0 (mod 16) and p(49n+14) ≡ 0 (mod 16) for n ≥ 0. We also find four congruence relations: p(4n) ≡ (−1) n p(n) (mod 16) for n ≥ 0, p(4n) ≡ (−1) n p(n) (mod 32) for n being not a square of an odd positive integer, p(4n) ≡ (−1) n p(n) (mod 64) for n ≡ 1, 2, 5 (mod 8) and p(4n) ≡ (−1) n p(n) (mod 128) for n ≡ 0 (mod 4).

International Journal of Trendy Research in Engineering and Technology, 2024

In this paper, we will be covering the background of modular forms and explain how they are used to prove many important results about partition function congruence. We first go over the background of molecular forms mainly focusing on modular forms over SL 2 (z). In addition to this, we will also introduce more general modular forms and half-integral weight molecular forms by Ramanujan's congruence proof. Apart from this, an additional family of Congruences was proven using the theory of modular forms over SL₂(z) and also using some computer-checkable computation, we will also modify the Method of ono to give a more general framework of proof. It allows us to straightforwardly extend the congruence on proving modules 13.17, 19, and 23 to give similar congruence to modules 29 and 31.

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 2014

In the case of Siegel modular forms of degree n, we prove that, for almost all prime ideals p in any ring of algebraic integers, mod 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.

Journal of Mathematical Analysis and Applications, 2012

In this paper, we give an alternative proof of four of Ramanujan's modular equations of degree 35, which have been proved by B.C. Berndt using the theory of modular forms. Our proofs involve only the identities stated by Ramanujan.

The Ramanujan Journal, 2013

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.