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A note on the distribution of normalized prime gaps

Volume 184 / 2018

János Pintz Acta Arithmetica 184 (2018), 413-418 MSC: 11N05, 11N36. DOI: 10.4064/aa171127-15-8 Published online: 18 September 2018

Abstract

Erdős conjectured in 1955 that if we normalize the sequence of prime gaps by dividing the individual gaps by the natural logarithm of the (say, smaller) prime then the resulting sequence is everywhere dense within the set of positive reals. Although there seemed to be no possibility to specify any concrete limit points, Erdős and Ricci independently proved more than 60 years ago that the set of limit points has a positive Lebesgue measure. Using the new method of Maynard and Tao and further many other ideas, Banks, Freiberg and Maynard showed that the set of limit points contains at least $T(1+o(1))/8$ limit points below $T$. In the present work it is proved by a modification of the above method that the same assertion remains true if we substitute 8 by 4 in the denominator.

Authors

  • János PintzRényi Mathematical Institute
    of the Hungarian Academy of Sciences
    Reáltanoda u. 13–15
    H-1053 Budapest, Hungary
    e-mail

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