Flows of Mellin transforms with periodic integrator
Journal de Théorie des Nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 455-469.

We study Mellin transforms N ^(s)= 1- 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 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 ^} λ1 which converges locally uniformly to ζ(s) as λ1, and show that they exhibit features similar to ζ(s). For example, N λ ^(s) has roughly T 2πlogT 2π-T 2π zeros in the critical strip up to height T and an infinite number of negative zeros, roughly at the points λ-1-2n (n).

Nous étudions les tranformées de Mellin N ^(s)= 1- x -s dN(x) pour lesquelles N(x)-x est périodique de période 1 dans le but d’examiner les “flots” de telles fonctions vers la fonction ζ(s) de Riemann et la possibilité de prouver l’hypothèse de Riemann avec cette approche. Nous montrons que, à part le cas trivial N(x)=x, la borne supérieure des parties réelles des zéros de n’importe quelle telle fonction est au moins 1 2.

Nous examinons un flot particulier de telles fonctions {N λ ^} λ1 qui converge localement uniformément vers ζ(s) quand λ1, et montrons qu’elles présentent un aspect similaire à ζ(s). Par exemple, N λ ^(s) a à peu près T 2πlogT 2π-T 2π zéros dans la bande critique jusqu’à la hauteur T, et une infinité de zéros négatifs, environ aux points λ-1-2n (n).

Received:
Published online:
DOI: 10.5802/jtnb.771
Classification: 11M41,  30C15
Keywords: Zeros of Mellin transforms, Lindelöf function
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     title = {Flows of {Mellin} transforms with periodic integrator},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
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Titus Hilberdink. Flows of Mellin transforms with periodic integrator. Journal de Théorie des Nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 455-469. doi : 10.5802/jtnb.771. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.771/

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