Toy Models for D. H. Lehmer’s Conjecture II

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.

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A SHORT NOTE ON THE MAHLER MEASURE AND LEHMER'S CONJECTURE

Journal of Pure and Applied Mathematics Advances and Applications ◽

10.18642/jpamaa_7100121768 ◽

2017 ◽

Vol 17 (1) ◽

pp. 47-60

Author(s):

Ali H. Hakami

Keyword(s):

Short Note ◽

Mahler Measure ◽

Lehmer's Conjecture

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A variant of Lehmer's conjecture

Journal of Number Theory ◽

10.1016/j.jnt.2006.06.004 ◽

2007 ◽

Vol 123 (1) ◽

pp. 80-91 ◽

Cited By ~ 3

Author(s):

V. Kumar Murty

Keyword(s):

Lehmer's Conjecture

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Lehmer's Conjecture for Hermitian Matrices over the Eisenstein and Gaussian Integers

The Electronic Journal of Combinatorics ◽

10.37236/2834 ◽

2013 ◽

Vol 20 (1) ◽

Author(s):

Graeme Taylor ◽

Gary Greaves

Keyword(s):

Mahler Measure ◽

Hermitian Matrices ◽

Gaussian Integers ◽

Lehmer's Conjecture ◽

Lehmer’S Problem

We solve Lehmer's problem for a class of polynomials arising from Hermitian matrices over the Eisenstein and Gaussian integers, that is, we show that all such polynomials have Mahler measure at least Lehmer's number $\tau_0 = 1.17628\dots$.

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The explicit Sato–Tate conjecture for primes in arithmetic progressions

International Journal of Number Theory ◽

10.1142/s179304212150069x ◽

2021 ◽

pp. 1-19

Author(s):

Trajan Hammonds ◽

Casimir Kothari ◽

Noah Luntzlara ◽

Steven J. Miller ◽

Jesse Thorner ◽

...

Keyword(s):

Modular Form ◽

Substantial Improvement ◽

Arithmetic Progressions ◽

Implied Constant ◽

Symmetric Power ◽

Tau Function ◽

Tate Conjecture ◽

Primes In Arithmetic Progressions ◽

Lehmer's Conjecture ◽

Level 1

Let [Formula: see text] be Ramanujan’s tau function, defined by the discriminant modular form [Formula: see text] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer’s conjecture asserts that [Formula: see text] for all [Formula: see text]; since [Formula: see text] is multiplicative, it suffices to study primes [Formula: see text] for which [Formula: see text] might possibly be zero. Assuming standard conjectures for the twisted symmetric power [Formula: see text]-functions associated to [Formula: see text] (including GRH), we prove that if [Formula: see text], then [Formula: see text] a substantial improvement on the implied constant in previous work. To achieve this, under the same hypotheses, we prove an explicit version of the Sato–Tate conjecture for primes in arithmetic progressions.

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A Variant of Lehmer’s Conjecture, II: The CM-case

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-002-4 ◽

2011 ◽

Vol 63 (2) ◽

pp. 298-326 ◽

Cited By ~ 3

Author(s):

Sanoli Gun ◽

V. Kumar Murty

Keyword(s):

Asymptotic Formula ◽

Fourier Coefficient ◽

Fourier Coefficients ◽

Complex Multiplication ◽

Rational Integer ◽

Know How ◽

Lehmer's Conjecture ◽

Hecke Eigenform

Abstract Let f be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer n has a factor common with the n-th Fourier coefficient of f. It has been shown in previous papers that this happens very often. In this paper, we give an asymptotic formula for the number of integers n for which (n, a(n)) = 1, where a(n) is the n-th Fourier coefficient of a normalized Hecke eigenform f of weight 2 with rational integer Fourier coefficients and having complex multiplication.

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