# Flows of Mellin transforms with periodic integrator

Titus Hilberdink^{[1]}

- [1] Department of Mathematics University of Reading Whiteknights PO Box 220 Reading RG6 6AX, UK

Journal de Théorie des Nombres de Bordeaux (2011)

- Volume: 23, Issue: 2, page 455-469
- ISSN: 1246-7405

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topHilberdink, Titus. "Flows of Mellin transforms with periodic integrator." Journal de Théorie des Nombres de Bordeaux 23.2 (2011): 455-469. <http://eudml.org/doc/219810>.

@article{Hilberdink2011,

abstract = {We study Mellin transforms $\hat\{N\}(s)=\int _\{1-\}^\{\infty \} x^\{-s\} dN(x)$ for which $N(x)-x$ is periodic with period $1$ in order to investigate ‘flows’ of such functions to Riemann’s $\zeta (s)$ and the possibility of proving the Riemann Hypothesis with such an approach. We show that, excepting the trivial case where $N(x)=x$, the supremum of the real parts of the zeros of any such function is at least $\frac\{1\}\{2\}$.We investigate a particular flow of such functions $\lbrace \hat\{N_\{\lambda \}\}\rbrace _\{\lambda \ge 1\}$ which converges locally uniformly to $\zeta (s)$ as $\lambda \rightarrow 1$, and show that they exhibit features similar to $\zeta (s)$. For example, $\hat\{N_\{\lambda \}\}(s)$ has roughly $\frac\{T\}\{2\pi \}\log \frac\{T\}\{2\pi \}-\frac\{T\}\{2\pi \}$ zeros in the critical strip up to height $T$ and an infinite number of negative zeros, roughly at the points $\lambda -1-2n$$(n\in \mathbb\{N\})$.},

affiliation = {Department of Mathematics University of Reading Whiteknights PO Box 220 Reading RG6 6AX, UK},

author = {Hilberdink, Titus},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {Zeros of Mellin transforms; Lindelöf function; zeros of Mellin transforms},

language = {eng},

month = {6},

number = {2},

pages = {455-469},

publisher = {Société Arithmétique de Bordeaux},

title = {Flows of Mellin transforms with periodic integrator},

url = {http://eudml.org/doc/219810},

volume = {23},

year = {2011},

}

TY - JOUR

AU - Hilberdink, Titus

TI - Flows of Mellin transforms with periodic integrator

JO - Journal de Théorie des Nombres de Bordeaux

DA - 2011/6//

PB - Société Arithmétique de Bordeaux

VL - 23

IS - 2

SP - 455

EP - 469

AB - We study Mellin transforms $\hat{N}(s)=\int _{1-}^{\infty } x^{-s} dN(x)$ for which $N(x)-x$ is periodic with period $1$ in order to investigate ‘flows’ of such functions to Riemann’s $\zeta (s)$ and the possibility of proving the Riemann Hypothesis with such an approach. We show that, excepting the trivial case where $N(x)=x$, the supremum of the real parts of the zeros of any such function is at least $\frac{1}{2}$.We investigate a particular flow of such functions $\lbrace \hat{N_{\lambda }}\rbrace _{\lambda \ge 1}$ which converges locally uniformly to $\zeta (s)$ as $\lambda \rightarrow 1$, and show that they exhibit features similar to $\zeta (s)$. For example, $\hat{N_{\lambda }}(s)$ has roughly $\frac{T}{2\pi }\log \frac{T}{2\pi }-\frac{T}{2\pi }$ zeros in the critical strip up to height $T$ and an infinite number of negative zeros, roughly at the points $\lambda -1-2n$$(n\in \mathbb{N})$.

LA - eng

KW - Zeros of Mellin transforms; Lindelöf function; zeros of Mellin transforms

UR - http://eudml.org/doc/219810

ER -

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