Results 101 to 110 of about 891 (145)

Computing the Ramanujan tau function

The Ramanujan Journal, 2006
We show that the Ramanujan tau function τ(n) can be computed by a randomized algorithm that runs in time \(n^{\frac{1}{2}+\varepsilon}\) for every O(\(n^{\frac{3}{4}+\varepsilon}\)) assuming the Generalized Riemann Hypothesis. The same method also yields a deterministic algorithm that runs in time O(\(n^{\frac{3}{4}+\varepsilon}\)) (without any ...
Denis Charles
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Sign Changes of the Ramanujan $$\tau $$-Function

, 2020
The signs and vanishing of Fourier coefficients of modular forms are important properties of modular forms and are closely related. The focus of this paper is on the coefficients of the powers of the Dedekind \(\eta \)-function, in particular the discriminant function \(\Delta \) and the Ramanujan \(\tau \)-function.
Bernhard Heim, Markus Neuhauser
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NONVANISHING OF THE RAMANUJAN TAU FUNCTION IN SHORT INTERVALS

International Journal of Number Theory, 2005
We provide new estimates for the gap function of the Delta function and for the number of nonzero values of the Ramanujan tau function in short intervals.
Emré Alkan, Alexandru Zaharescu
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Historical Remark on Ramanujan′s Tau Function

The American Mathematical Monthly, 2015
It is shown that Ramanujan could have proved a special case of his conjecture that his tau function is multiplicative without any recourse to modularity results.
Kenneth S. Williams
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Odd prime values of the Ramanujan tau function

The Ramanujan Journal, 2013
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Nik Lygerōs, Olivier Rozier
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On values of Ramanujan’s tau function involving two prime factors

The Ramanujan Journal, 2023
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Wenwen Lin, Wenjun Ma
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A Heat Kernel Associated to Ramanujan's Tau Function

The Ramanujan Journal, 2000
The main result in this paper is an identity for sums of the form \(\sum _{n=1}^{\infty} \tau ^2(n)g(n)\), where \(\tau (n)\) is the Ramanujan function and \(g\) is a sufficiently nice function. In the special case \(g(x)= e^{-yx}\) with positive \(y\) tending to zero, the sum in question equals \[ 12\Gamma (11)y^{-12}+ y^{-11}\sum_{\rho }y^{-\rho /2 ...
Hafner, James Lee, Stopple, Jeffrey
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Ramanujan’s Tau-Function in Terms of Bell Polynomials

Indian Journal of Advanced Mathematics, 2023
We obtain a recurrence relation for the Ramanujan’s tau-function involving the sum of divisors function, and the solution of this recurrence gives a closed formula for 𝝉(𝒏) in terms of the complete Bell ...
Dr. R. Sivaraman, H. N. Núñez-Yépez, Prof. J. López-Bonilla   +2 more
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Ramanujan’s Tau Function

2017
Ramanujan’s tau function satisfies a recurrence for each prime p. Elementary proofs for \(p = 2, 3, 5\) and 7, using identities obtained in earlier chapters, are given.
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A central limit theorem for Ramanujan’s tau function

The Ramanujan Journal, 2012
Let \(\tau\) be the Ramanujan tau function. By definition, \(\tau(n)\in \mathbb{Z}\) and Lehmer (1947) conjectured that \(\tau(n)=0\) cannot happen. By Deligne's work, now it is known that \(|\tau(n)|/n^{11/2}\leq d(n)\) where \(d(n)\) counts the number of divisors of \(n\).
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