Let p n denote the nth smallest prime number, and let L denote the set of limit points of the sequence tpp n`1´pn q{ log p n u 8 n"1 of normalized differences between consecutive primes. We show that for k " 9 and for any sequence of k nonnegative real numbers β 1 ď β 2 﨨¨ď β k , at least one of the numbers β j´βi (1 ď i ă j ď k) belongs to L. It follows that at least 12.5% of all nonnegative real numbers belong to L.
In a stunning new advance towards the Prime k-tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple Hpxq " tgx`h j u k j"1 of linear forms in Zrxs, the set Hpnq " tgn`h j u k j"1 contains at least m primes for infinitely many n P N. In this note, we deduce that Hpnq " tgn`h j u k j"1 contains at least m consecutive primes for infinitely many n P N. We answer an old question of Erdős and Turán by producing strings of m`1 consecutive primes whose successive gaps δ 1 , . . . , δ m form an increasing (resp. decreasing) sequence. We also show that such strings exist with δ j´1 | δ j for 2 ď j ď m. For any coprime integers a and D we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class a mod D.
A. A well-known conjecture asserts that, for any given positive real number λ and nonnegative integer m, the proportion of positive integers n ď x for which the interval pn, n`λ log ns contains exactly m primes is asymptotically equal to λ m e´λ{m! as x tends to infinity. We show that the number of such n is at least x 1´op1q .
A. We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors-Keating, and Smilansky, we formulate an analog of the Hardy-Littlewood prime k-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed. Consequently, by work of Rudnick and Ueberschär, the level spacings of arithmetic toral point scatterers, in the weak coupling limit, are also Poisson distributed. We also give numerical evidence for the conjecture and its implications.
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