An algorithmic approach to Ramanujan’s congruences (original) (raw)
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Rogers-Ramanujan Functions, Modular Functions, and Computer Algebra
2016
Many generating functions for partitions of numbers are strongly related to modular functions. This article introduces such connections using the Rogers-Ramanujan functions as key players. After exemplifying basic notions of partition theory and modular functions in tutorial manner, relations of modular functions to q-holonomic functions and sequences are discussed. Special emphasis is put on supplementing the ideas presented with concrete computer algebra. Despite intended as a tutorial, owing to the algorithmic focus the presentation might contain aspects of interest also to the expert. One major application concerns an algorithmic derivation of Felix Klein’s classical icosahedral equation.
A simple proof of the Ramanujan conjecture for powers of 5
Journal für die reine und angewandte Mathematik (Crelles Journal), 2000
Ramanujan conjectured, and G. N. Watson proved, that if n is of a specific form then p(n), the number of partitions of n, is divisible by a high power of 5. In the present note, we establish appropriate generating function formulae, from which the truth of Ramanujan's conjecture, as well as some results of a similar type due to Watson, are shown to follow easily. Furthermore, we derive two new congruences for the partition function. Our proofs are more straightforward than those of Watson and more recent writers and use only classical identities of Euler and Jacobi. 1. Ramanujan [5] conjectured, and G. N. Watson [6] proved, that if α ≥ 1 and if δ α is the reciprocal modulo 5 α of 24, then (1.1) p(5 α n + δ α) ≡ 0 (mod 5 α). Watson also proved that if α is odd and at least 3, then (1.2) p(5 α n + δ α ± 5 α−1) ≡ 0 (mod 5 α). Watson's proof of (1.1) has subsequently been simplified by A. O. L. Atkin [2], M. I. Knopp [4] and G. E. Andrews [1]. However, whereas all these writers find it necessary to rely on the modular equation of fifth order, we do not. Instead, our proofs of (1.1) and (1.2) depend on an elementary lemma ((2.7) below). Furthermore, our main result, stated below, enables us to calculate explicitly the generating functions for p(5 α n + δ α). We carry out these calculations for α = 1, 2, 3, 4, and hence are able to prove the new congruence relations (1.3) p(125n + 99) ≡ 89 × 25p(5n + 4) (mod 5 6)
The Ramanujan Journal, 2009
We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x 2 + 5y 2 . Making use of Ramanujan's 1 ψ 1 summation formula we establish a new Lambert series identity for P ∞ n,m=−∞ q n 2 +5m 2 . Conjectures of Fermat and Euler are shown to follow easily from this new formula. But we don't stop there. Employing various formulas found in Ramanujan's notebooks and using a bit of ingenuity we obtain a collection of new Lambert series for certain infinite products associated with quadratic forms such as x 2 + 6y 2 , 2x 2 + 3y 2 , x 2 + 15y 2 , 3x 2 + 5y 2 , x 2 + 27y 2 , x 2 + 5(y 2 + z 2 + w 2 ), 5x 2 + y 2 + z 2 + w 2 . In the process, we find many new multiplicative eta-quotients and determine their coefficients.
Ramanujan-type congruences for overpartitions modulo 5
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).
Ramanujan-style congruences for prime level
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.
A survey of results on a generalization of Ramanujan sum
Ramanujan sum is a sum of powers of primitive roots of unity defined by Srinivasa Ramanujan. After this, there have been many generalizations of the Ramanujan sum one of which was given by E. Cohen. In a series of articles, he proved that several interesting properties of the classical Ramanujan sum extends to his generalization as well. Many other authors followed the footsteps of Cohen to give various such generalized results. In this survey article, we list some of the most important properties of the original sum and the generalization and also give some expected results using the generalized sum
New modular relations for Ramanujan's parameter mu(q)\mu(q)mu(q)
In his ‘lost’ notebook, S. Ramanujan introduced the parameter � \mu(q) := R(q)R(q^4) related to the Rogers-Ramanujan continued fraction R(q). In this paper, we establish some new P -Q modular equations of degree 5. We establish some general formulas for the explicit evaluations of the ratios of Ramanujan’s theta function \varphi. We obtain several new modular relations connecting � \mu(q) with � \mu(q^n) for different positive integer n > 1; reciprocity theorems and also compute several new explicit evaluations.
Ramanujan Congruences for a Class of Eta Quotients
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.