En supposant l’hypothèse de Riemann, on examine la distribution d’écarts entre les zéros du . On démontre qu’une proportion positive d’écarts sont inférieurs à fois l’écart moyen et que dans l’autre direction, une proportion positive d’écarts sont fois supérieurs à l’écart moyen. On montre également l’existence d’un nombre infini d’écarts normalisés qui sont inférieurs (supérieurs) à (respectivement ).
Assuming the Riemann hypothesis, we investigate the distribution of gaps between the zeros of . We prove that a positive proportion of gaps are less than times the average spacing and, in the other direction, a positive proportion of gaps are greater than times the average spacing. We also exhibit the existence of infinitely many normalized gaps smaller (larger) than (, respectively).
@article{JTNB_2010__22_2_287_0, author = {Hung Manh Bui}, title = {Gaps between zeros of the derivative of the {Riemann} $\xi $-function}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {287--305}, publisher = {Universit\'e Bordeaux 1}, volume = {22}, number = {2}, year = {2010}, doi = {10.5802/jtnb.716}, mrnumber = {2769063}, zbl = {1223.11103}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.716/} }
TY - JOUR AU - Hung Manh Bui TI - Gaps between zeros of the derivative of the Riemann $\xi $-function JO - Journal de théorie des nombres de Bordeaux PY - 2010 SP - 287 EP - 305 VL - 22 IS - 2 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.716/ DO - 10.5802/jtnb.716 LA - en ID - JTNB_2010__22_2_287_0 ER -
%0 Journal Article %A Hung Manh Bui %T Gaps between zeros of the derivative of the Riemann $\xi $-function %J Journal de théorie des nombres de Bordeaux %D 2010 %P 287-305 %V 22 %N 2 %I Université Bordeaux 1 %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.716/ %R 10.5802/jtnb.716 %G en %F JTNB_2010__22_2_287_0
Hung Manh Bui. Gaps between zeros of the derivative of the Riemann $\xi $-function. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 2, pp. 287-305. doi : 10.5802/jtnb.716. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.716/
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