scholarly journals On the number of representations of certain quadratic forms and a formula for the Ramanujan Tau function

2018 ◽  
Vol 58 (2) ◽  
pp. 233-244
Author(s):  
B. Ramakrishnan ◽  
Brundaban Sahu ◽  
Anup Kumar Singh
Keyword(s):  
Quadratic Forms ◽  
Tau Function ◽  
Ramanujan Tau Function

NONVANISHING OF THE RAMANUJAN TAU FUNCTION IN SHORT INTERVALS

2005 ◽  
Vol 01 (01) ◽  
pp. 45-51 ◽  
Author(s):  
EMRE ALKAN ◽  
ALEXANDRU ZAHARESCU
Keyword(s):  
Delta Function ◽  
Gap Function ◽  
Tau Function ◽  
Short Intervals ◽  
Ramanujan Tau Function

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.

scholarly journals Odd values of the Ramanujan $\tau$-function

1987 ◽  
Vol 79 ◽  
pp. 391-395 ◽  
Author(s):  
M.Ram Murty ◽  
V.Kumar Murty ◽  
T.N. Shorey
Keyword(s):  
Tau Function ◽  
Ramanujan Tau Function

scholarly journals Wigner Distribution Analysis Applied to Lehmer’s Conjecture on the Ramanujan tau Function

2021 ◽  
pp. 19-28
Author(s):  
Takaaki Musha
Keyword(s):  
Signal Processing ◽  
Dirichlet Series ◽  
Wigner Distribution ◽  
Distribution Analysis ◽  
Euler Product ◽  
Instantaneous Spectrum ◽  
Tau Function ◽  
Lehmer's Conjecture ◽  
Ramanujan Tau Function

Wigner distribution is a tool for signal processing to obtain instantaneous spectrum of a signal. By using Wigner distribution analysis, another representation of the Euler product can be obtained for Dirichlet series of the Ramanujan tau function. From which, it can be proved that the Ramanujan tau function never become zero for all numbers.

Odd prime values of the Ramanujan tau function

2013 ◽  
Vol 32 (2) ◽  
pp. 269-280 ◽  
Author(s):  
Nik Lygeros ◽  
Olivier Rozier
Keyword(s):  
Tau Function ◽  
Ramanujan Tau Function

Computing the Ramanujan tau function

2006 ◽  
Vol 11 (2) ◽  
pp. 221-224 ◽  
Author(s):  
Denis Xavier Charles
Keyword(s):  
Tau Function ◽  
Ramanujan Tau Function

scholarly journals Odd values of the Ramanujan tau function

2021 ◽  
Author(s):  
Michael A. Bennett ◽  
Adela Gherga ◽  
Vandita Patel ◽  
Samir Siksek
Keyword(s):  
Tau Function ◽  
Ramanujan Tau Function

Waring's problem with the Ramanujan $ \tau$-function

2008 ◽  
Vol 72 (1) ◽  
pp. 35-46 ◽  
Author(s):  
M Z Garaev ◽  
V C Garcia ◽  
S V Konyagin
Keyword(s):  
Tau Function ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Ramanujan Tau Function

Computing coefficients of modular forms

2011 ◽  
Author(s):  
Bas Edixhoven
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Galois Representations ◽  
Theta Functions ◽  
Positive Definite ◽  
Hecke Operators ◽  
Linear Combinations ◽  
Tau Function ◽  
Arbitrary Weight

This chapter applies the main result on the computation of Galois representations attached to modular forms of level one to the computation of coefficients of modular forms. It treats the case of the discriminant modular form, that is, the computation of Ramanujan's tau-function at primes, and then deals with the more general case of forms of level one and arbitrary weight k, reformulated as the computation of Hecke operators Tⁿ as ℤ-linear combinations of the Tᵢ with i < k = 12. The chapter gives an application to theta functions of even, unimodular positive definite quadratic forms over ℤ.

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