Results 11 to 20 of about 891 (145)
On a function related to Ramanujan′s Tau function [PDF]
International Journal of Mathematics and Mathematical Sciences, 1985 For the function ψ = ψ12, defined by , the author derives two simple formulas. The simpler of these two formulas is expressed solely in terms of the well‐known sum‐of‐divisors function.
John A. Ewell
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Odd values of the Ramanujan tau function [PDF]
Mathematische Annalen, 2021 We prove a number of results regarding odd values of the Ramanujan $τ$-function. For example, we prove the existence of an effectively computable positive constant $κ$ such that if $τ(n)$ is odd and $n \ge 25$ then either \[ P(τ(n)) \; > \; κ\cdot \frac{\log\log\log{n}}{\log\log\log\log{n}} \] or there exists a prime $p \mid n$ with $τ(p)=0$.
Michael A. Bennett +3 more
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Odd values of the Ramanujan $\tau$-function [PDF]
Bulletin de la Société mathématique de France, 1987 This short but very beautiful paper contains the first step towards a solution of the deep conjecture of Atkin and Serre that \(| \tau(p)| \gg_{\varepsilon} p^{(9/2)-\varepsilon}\) (for every positive \(\varepsilon\) and prime \(p)\) where \(\tau(p)\) is the \(p\)th Fourier coefficient of the unique normalized cusp form of weight \(12\) for the full ...
M. Ram Murty +2 more
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Even Values of Ramanujan’s Tau-Function [PDF]
La Matematica, 2021 In the spirit of Lehmer's speculation that Ramanujan's tau-function never vanishes, it is natural to ask whether any given integer $ $ is a value of $ (n)$. For odd $ $, Murty, Murty, and Shorey proved that $ (n)\neq $ for sufficiently large $n$. Several recent papers have identified explicit examples of odd $ $ which are not tau-values. Here we
Balakrishnan, Jennifer +2 more
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JOURNAL OF RAMANUJAN SOCIETY OF MATHEMATICS AND MATHEMATICAL SCIENCES, 2023 We exhibit a recurrence relation for the Ramanujan’s tau-function involving the sum of divisors function, whose solution gives a closed formula for τ(n) in terms of complete Bell polynomials. Besides, we show that it is possible to write τ(n) in terms of the compositions of n.
López-Bonilla, J. +3 more
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New Formulas for the Ramanujan Tau Function
The Ramanujan tau function is the Fourier coefficient of the discriminant modular form. We obtain some new formulas for the Ramanujan tau function.
Huan Xiao
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Combinatorial interpretations of Ramanujan’s tau function [PDF]
Discrete Mathematics, 2018 We use a q-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan's tau function which involves t-cores and a new class of partitions which we call (m,k)-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.
Frank Garvan, Michael J. Schlosser
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Shifted convolution of divisor function $d_3$ and Ramanujan $\tau$ function [PDF]
, 2013 Comment: This short note is based on my paper arXiv:1202.1157.
Ritabrata Munshi
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Variations of Lehmer's Conjecture for Ramanujan's tau-function
Journal of Number Theory, 2022 We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for $n>1$ we prove that $$τ(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 691\}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of ...
Balakrishnan, Jennifer S. +2 more
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A Formula for Ramanujan's Tau Function [PDF]
Proceedings of the American Mathematical Society, 1984 A formula for Ramanujan’s tau function τ \tau , defined by ∑ 1 ∞ τ ( n ) x n = x ∏ 1 ∞ ( 1 −
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