A note on the distribution of normalized prime gaps

János Pintz¹

¹ Rényi Mathematical Institute of the Hungarian Academy of Sciences Reáltanoda u. 13–15 H-1053 Budapest, Hungary

Acta Arithmetica, Tome 184 (2018) no. 4, pp. 413-418.

Résumé

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.

Détail

DOI : 10.4064/aa171127-15-8

Keywords: erd conjectured normalize sequence prime gaps dividing individual gaps natural logarithm say smaller prime resulting sequence everywhere dense within set positive reals although there seemed possibility specify concrete limit points erd ricci independently proved nbsp years ago set limit points has positive lebesgue measure using method maynard tao further many other ideas banks freiberg maynard showed set limit points contains least limit points below nbsp present work proved modification above method assertion remains substitute nbsp denominator