scholarly journals Partial sums of random multiplicative functions and extreme values of a model for the Riemann zeta function

2020 ◽  
Author(s):  
Marco Aymone ◽  
Winston Heap ◽  
Jing Zhao
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Extreme Values ◽  
Partial Sums ◽  
Multiplicative Functions ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals On The Tails of the Exponential Series

1994 ◽  
Vol 37 (2) ◽  
pp. 278-286 ◽  
Author(s):  
C. Yalçin Yildirim
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Partial Sums ◽  
Maclaurin Series ◽  
Exponential Series ◽  
Asymptotic Estimation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

AbstractA relation between the zeros of the partial sums and the zeros of the corresponding tails of the Maclaurin series for ez is established. This allows an asymptotic estimation of a quantity which came up in the theory of the Riemann zeta-function. Some new properties of the tails of ez are also provided.

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

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