Gaps between zeros of the derivative of the Riemann $\xi$-function

Bui, Hung Manh

Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 287-305.

Assuming the Riemann hypothesis, we investigate the distribution of gaps between the zeros of ${\xi }^{\prime }\left(s\right)$. We prove that a positive proportion of gaps are less than $0.796$ times the average spacing and, in the other direction, a positive proportion of gaps are greater than $1.18$ times the average spacing. We also exhibit the existence of infinitely many normalized gaps smaller (larger) than $0.7203$ ($1.5$, respectively).

En supposant l’hypothèse de Riemann, on examine la distribution d’écarts entre les zéros du ${\xi }^{\prime }\left(s\right)$. On démontre qu’une proportion positive d’écarts sont inférieurs à $0.796$ fois l’écart moyen et que dans l’autre direction, une proportion positive d’écarts sont $1.18$ fois supérieurs à l’écart moyen. On montre également l’existence d’un nombre infini d’écarts normalisés qui sont inférieurs (supérieurs) à $0.7203$ (respectivement $1.5$).

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