The distribution of Euler–Kronecker constants of quadratic fields

Lamzouri, Youness

doi:10.1016/j.jmaa.2015.06.065

Cited by 18 publications

(10 citation statements)

References 12 publications

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“…Additionally, we also notice the coefficient C 1 which appears in all of the theorems describing the behaviour of Φ X (τ )( cf. [12,7,18]). We find over function fields that the coefficient C 1 is no longer fixed, but remains bounded between − ln(cosh(q))/q+tanh(q) and 1/ ln(q) − ln(cosh(c))/c + tanh(c).…”

Section: P |+1mentioning

confidence: 99%

Complex Moments and the Distribution of Values of Over Function Fields With Applications to Class Numbers

Lumley

2018

Mathematika

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In this paper we investigate the moments and the distribution of L(1, χ D ), where χ D varies over quadratic characters associated to square-free polynomials D of degree n over F q , as n → ∞. Our first result gives asymptotic formulas for the complex moments of L(1, χ D ) in a large uniform range. Previously, only the first moment has been computed due to work of Andrade and Jung. Using our asymptotic formulas together with the saddle-point method, we show that the distribution function of L(1, χ D ) is very close to that of a corresponding probabilistic model. In particular, we uncover an interesting feature in the distribution of large (and small) values of L(1, χ D ), that is not present in the number field setting. We also obtain Ω-results for the extreme values of L(1, χ D ), which we conjecture to be best possible. Specializing n = 2g + 1 and making use of one case of Artin's class number formula, we obtain similar results for the class number h D associated to F q (T )[ √ D]. Similarly, specializing to n = 2g + 2 we can appeal to the second case of Artin's class number formula and deduce analogous results for h

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Section: P |+1mentioning

confidence: 99%

Complex Moments and the Distribution of Values of Over Function Fields With Applications to Class Numbers

Lumley

2018

Mathematika

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“…which also can be deduced from (24). Moreover, logarithmic differentiation of the L-function factorization from (26) yields…”

Section: 3mentioning

confidence: 97%

Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms

Ciolan¹,

Languasco²,

Moree³

2021

Preprint

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In 1961, Rankin determined the asymptotic behavior of the number S k,q (x) of positive integers n ≤ x for which a given prime q does not divide σ k (n), the k-th divisor sum function. By computing the associated Euler-Kronecker constant γ k,q , which depends on the arithmetic of certain subfields of Q(ζ q ), we obtain the second order term in the asymptotic expansion of S k,q (x). Using a method developed by Ford, Luca and Moree (2014), we determine the pairs (k, q) with (k, q − 1) = 1 for which Ramanujan's approximation to S k,q (x) is better than Landau's. This entails checking whether γ k,q < 1/2 or not, and requires a substantial computational number theoretic input and extensive computer usage. We apply our results to study the non-divisibility of Fourier coefficients of six cusp forms by certain exceptional primes, extending the earlier work of Moree ( 2004), who disproved several claims made by Ramanujan on the non-divisibility of the Ramanujan tau function by five such exceptional primes.

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“…Such data are in agreement with the estimate proved by Ihara-Murty-Shimura [12] (please remark that our M q is denoted as Q m there) since they proved that M q ≤ (2 + o (1)) log log q as q tends to infinity, under the assumption of the Generalised Riemann Hypothesis. On the other hand, Lamzouri, in a personal communication with the first author, remarked that, by adapting the techniques in his paper [16], one can show that if q is a large prime then M q ≥ (1 + o (1)) log log q.…”

Section: Extremal Values Of L /L(1 χ)mentioning

confidence: 99%

A fast algorithm to compute the Ramanujan-Deninger gamma-function and some number-theoretic applications

Languasco,

Righi

2020

Preprint

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A. We introduce a fast algorithm to compute the Ramanujan-Deninger gamma function and its logarithmic derivative at positive values. Such an algorithm allows us to greatly extend the numerical investigations about the Euler-Kronecker constants G q , G + q and M q = max χ χ 0 |L /L(1, χ)|, where q is an odd prime, χ runs over the primitive Dirichlet characters mod q, χ 0 is the trivial Dirichlet character mod q and L(s, χ) is the Dirichlet L-function associated to χ. Using such algorithms we obtained that G 50040955631 = −0.16595399 . . . and G + 50040955631 = 13.89764738 . . . thus getting a new negative value for G q . Moreover we also computed G q , G + q and M q for every odd prime q, 10 6 < q ≤ 10 7 , thus extending the results in [17]. As a consequence we obtain that both G q and G + q are positive for every odd prime q up to 10 7 and that 17 20 log log q < M q < 5 4 log log q for every odd prime 1531 < q ≤ 10 7 . In fact the lower bound holds true for q > 13. The programs used and the results here described are collected at the following address http://www.math.unipd.it/ ~languasc/Scomp-appl.html.

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The distribution of Euler–Kronecker constants of quadratic fields

Cited by 18 publications

References 12 publications

Complex Moments and the Distribution of Values of Over Function Fields With Applications to Class Numbers

Complex Moments and the Distribution of Values of Over Function Fields With Applications to Class Numbers

Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms

A fast algorithm to compute the Ramanujan-Deninger gamma-function and some number-theoretic applications

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