An example in Beurling's theory of generalised primes

Faez Al-Maamori; Titus Hilberdink

Displaying similar documents to “An example in Beurling's theory of generalised primes”

Ergodic Universality Theorems for the Riemann Zeta-Function and other $L$ -Functions

Jörn Steuding (2013)

Journal de Théorie des Nombres de Bordeaux

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We prove a new type of universality theorem for the Riemann zeta-function and other $L$ -functions (which are universal in the sense of Voronin’s theorem). In contrast to previous universality theorems for the zeta-function or its various generalizations, here the approximating shifts are taken from the orbit of an ergodic transformation on the real line.

Some infinite sums identities

Meher Jaban, Sinha Sneh Bala (2015)

Czechoslovak Mathematical Journal

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We find the sum of series of the form $\sum_{i = 1}^{\infty} \frac{f (i)}{i^{r}}$ for some special functions $f$ . The above series is a generalization of the Riemann zeta function. In particular, we take $f$ as some values of Hurwitz zeta functions, harmonic numbers, and combination of both. These generalize some of the results given in Mező’s paper (2013). We use multiple zeta theory to prove all results. The series sums we have obtained are in terms of Bernoulli numbers and powers of $π$ .

$Ω_{\pm}$ -results of the error term in the mean square formula of the Riemann zeta-function in the critical strip

Yuk-Kam Lau, Kai-Man Tsang (2001)

Acta Arithmetica

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Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line

Thomas Christ, Justas Kalpokas (2013)

Journal de Théorie des Nombres de Bordeaux

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We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments $I_{k, l} (T) = \int_{0}^{T} {| ζ^{(l)} (\frac{1}{2} + i t) |}^{2 k} d t$ , where $l$ is a non-negative integer and $k \geq 1$ a rational number. In particular, these lower bounds are of the expected order of magnitude for $I_{k, l} (T)$ .

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Hafedh Herichi, Michel L. Lapidus (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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We survey some of the universality properties of the Riemann zeta function $ζ (s)$ and then explain how to obtain a natural quantization of Voronin’s universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in [60]. It also makes an essential use of the functional analytic framework developed by the authors in [25] for rigorously studying the spectral operator...

The size of the Lerch zeta-function at places symmetric with respect to the line $ℜ (s) = 1 / 2$

Ramūnas Garunkštis, Andrius Grigutis (2019)

Czechoslovak Mathematical Journal

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Let $ζ (s)$ be the Riemann zeta-function. If $t \geq 6.8$ and $σ > 1 / 2$ , then it is known that the inequality $| ζ (1 - s) | > | ζ (s) |$ is valid except at the zeros of $ζ (s)$ . Here we investigate the Lerch zeta-function $L (λ, α, s)$ which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the critical line. However, for equal parameters $λ = α$ it is still possible to obtain a certain version of the inequality $| L (λ, λ, 1 - \bar{s}) | > | L (λ, λ, s) |$ .

Sur l'élimination des "infinis" en théorie quantique des champs : la régularisation zeta à l'épreuve de l'interprétation de Colombeau ou vice versa

Charpentier Éric

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Zeta regularization is one of the heuristic techniques used, specifically, in quantum field theory (QFT) in order to extract from divergent expressions some finite and well-defined values. Colombeau’s theory of generalized functions (containing the distributions and allowing their multiplication) provides a mathematically rigorous setting for QFT, where only convergent expressions appear. The aim of this paper is to compare these two points of view, both in their underlying logic and...

Dirichlet series induced by the Riemann zeta-function

Jun-ichi Tanaka (2008)

Studia Mathematica

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The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on $^{ω}$ , the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form $(a_{p}, s) = \prod_{p} {(1 - a_{p} p^{- s})}^{- 1}$ for $a_{p}$ in $^{ω}$ . Among other things, using the Haar measure on $^{ω}$ for measuring the asymptotic behavior of ζ(s) in the critical strip, we shall prove, in a weak sense, the mean-value theorem for ζ(s), equivalent to the Lindelöf hypothesis.