Large prime gaps and probabilistic models

Abstract

We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval [1, x]. Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set.

Cyclotomic polynomials with prescribed height and prime number theory

“…) . Further work on d n can be found in [4,11,16]. If Cramér's conjecture holds true, then the next lemma implies that N(x) = O(1).…”

“…The most famous one is Legendre's that there is a prime between consecutive squares is a bit weaker, but for example Firoozbakht's conjecture that p 1/n n is a strictly decreasing function of n is much stronger. Firoozbakht's conjecture implies that d n < (log p n ) 2 −log p n + 1 for all n sufficient large (see Sun [27]), contradicting a heuristic model, see Banks et al [4], suggesting that given any ǫ > 0 there are infinitely many n such that d n > (2e −γ − ǫ)(log p n ) 2 , with γ Euler's constant. A very classical conjecture of Cramér states that p n+1 − p n = O((log p n ) 2 ), which if true, clearly shows that the claimed bound in Conjecture 4 holds for all sufficiently large n.…”

Large gaps between primes in arithmetic progressions -- an empirical approach

“…Since the same sieving rules as used in the sieve of Eratosthenes apply to both the primes in general and primes in an arithmetic progression (AP), 1 it is reasonable to suggest that, by and large, results for primes in AP can be compared to statistical models like the ones that Granville [7] or Banks, Ford and Tao [1] use for the study of prime gaps.…”

Apie sveikųjų skaičių sekų, asocijuotų su pirminiais dvyniais, apskaičiavimą

“…Sena problema išlieka aktuali ir šiandien. Kasmet pasirodo jos, arba jos apibendrinimų, analiziniai [5,1,2] arba skaitiniai [6,8] tyrimai.…”