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The moments of central values of families of L-functions have recently attracted much attention and, with the work of Keating and Snaith [(2000) Commun. Math. Phys. 214, 57-89 and 91-110], there are now precise conjectures for their limiting values. We develop a simple method to establish lower bounds of the conjectured order of magnitude for several such(More)
Contrary to what would be predicted on the basis of Cramér's model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x + H) − Cramér [4] modeled the distribution of prime numbers by independent random variables X n (for n ≥ 3) that take the value 1 (n is " prime ") with probability 1/ log n and take the value 0 (n(More)
made a spectacular breakthrough in the study of prime numbers. Resolving a long-standing open problem, they proved that there are infinitely many primes for which the gap to the next prime is as small as we want compared to the average gap between consecutive primes. Before their work, it was only known that there were infinitely many gaps which were about(More)