Extreme values of the Riemann zeta function

1977 ◽  
Vol 52 (1) ◽  
pp. 511-518 ◽  
Author(s):  
Hugh L. Montgomery
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Extreme Values ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals Extreme values of the Riemann zeta function and its argument

2018 ◽  
Vol 372 (3-4) ◽  
pp. 999-1015 ◽  
Author(s):  
Andriy Bondarenko ◽  
Kristian Seip
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Extreme Values ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

On large values of L(σ,χ)

2018 ◽  
Vol 70 (3) ◽  
pp. 831-848 ◽  
Author(s):  
Christoph Aistleitner ◽  
Kamalakshya Mahatab ◽  
Marc Munsch ◽  
Alexandre Peyrot
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Probabilistic Models ◽  
Extreme Values ◽  
Resonance Method ◽  
Critical Strip ◽  
Principal Character ◽  
Large Order ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract In recent years, a variant of the resonance method was developed which allowed to obtain improved Ω-results for the Riemann zeta function along vertical lines in the critical strip. In the present paper, we show how this method can be adapted to prove the existence of large values of |L(σ,χ)| in the range σ∈(1/2,1], and to estimate the proportion of characters for which |L(σ,χ)| is of such a large order. More precisely, for every fixed σ∈(1/2,1), we show that for all sufficiently large q, there is a non-principal character χ(modq) such that log|L(σ,χ)|≥C(σ)(logq)1−σ(loglogq)−σ. In the case σ=1, we show that there is a non-principal character χ(modq) for which |L(1,χ)|≥eγ(log2q+log3q−C). In both cases, our results essentially match the prediction for the actual order of such extreme values, based on probabilistic models.

Number theory

2018 ◽  
Author(s):  
Thomas Spencer
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Random Matrix ◽  
Riemann Zeta Function ◽  
Matrix Theory ◽  
Extreme Values ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Eigenvalues Of Random Matrices ◽  
Areas Of Interest

This article examines some of the connections between random matrix theory (RMT) and number theory, including the modelling of the value distributions of the Riemann zeta function and other L-functions as well as the statistical distribution of their zeros. Number theory has been used in RMT to address seemingly disparate questions, such as modelling mean and extreme values of the Riemann zeta function and counting points on curves. One thing in common among the applications of RMT to number theory is the L-function. The statistics of the critical zeros of these functions are believed to be related to those of the eigenvalues of random matrices. The article first considers the truth of the generalized Riemann hypothesis before discussing the values of the Riemann zeta function, the values of L-functions, and further areas of interest with respect to the connections between RMT and number theory

scholarly journals LARGE VALUES OF L-FUNCTIONS ON THE 1-LINE

2020 ◽  
pp. 1-14
Author(s):  
ANUP B. DIXIT ◽  
KAMALAKSHYA MAHATAB
Keyword(s):  
Zeta Function ◽  
Lower Bounds ◽  
Riemann Zeta Function ◽  
Extreme Values ◽  
Zeta Functions ◽  
General Family ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We study lower bounds of a general family of L-functions on the $1$ -line. More precisely, we show that for any $F(s)$ in this family, there exist arbitrarily large t such that $F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$ , where m is the order of the pole of $F(s)$ at $s=1$ . This is a generalisation of the result of Aistleitner, Munsch and Mahatab [‘Extreme values of the Riemann zeta function on the $1$ -line’, Int. Math. Res. Not. IMRN2019(22) (2019), 6924–6932]. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type $L(s,f\times f)$ on the $1$ -line.

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