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

Abstract

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We study Mellin transforms N ^ ( s ) = ∫ 1 - ∞ x - s d N ( x ) for which N ( x ) - x is periodic with period 1 in order to investigate ‘flows’ of such functions to Riemann’s ζ ( 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 1 2 .We investigate a particular flow of such functions { N λ ^ } λ ≥ 1 which converges locally uniformly to ζ ( s ) as λ → 1 , and show that they exhibit features similar to ζ ( s ) . For example, N λ ^ ( s ) has roughly T 2 π log T 2 π - T 2 π zeros in the critical strip up to height T and an infinite number of negative zeros, roughly at the points λ - 1 - 2 n ( n ∈ ℕ ) .

How to cite

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Hilberdink, 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 -

References

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  1. T. M. Apostol, Introduction to Analytic Number Theory. Springer, 1976. Zbl1154.11300MR434929
  2. P. T. Bateman and H. G. Diamond, Analytic Number Theory. World Scientific Publishing, 2004. Zbl1074.11001MR2111739
  3. T. W. Hilberdink, A lower bound for the Lindelöf function associated to generalised integers. J. Number Theory 122 (2007), 336–341. Zbl1159.11036MR2292259
  4. T. W. Hilberdink and M. L. Lapidus, Beurling zeta functions, Generalised Primes, and Fractal Membranes. Acta Appl Math 94 (2006), 21–48. Zbl1133.11057MR2271675
  5. E. C. Titchmarsh, The Theory of Functions. Second edition, Oxford University Press, 1986. Zbl0336.30001MR882550
  6. E. C. Titchmarsh, The Theory of the Riemann Zeta-function. Second edition, Oxford University Press, 1986. Zbl0601.10026MR882550

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