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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)

3 Preface AG to work on: sort out / finalize? part 1. Sort out what we discuss about Halasz once the paper has been written. Ch3.3, 3.10 (Small gaps)and then all the Linnik stuff to be cleaned up; i.e. all of chapter 4. Sort out 5.6, 5.7 and chapter 6 ! Riemann's seminal 1860 memoir showed how questions on the distribution of prime numbers are more-or-less… (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)

- K. SOUNDARARAJAN, Dan Goldston, János Pintz, Cem Yıldırım
- 2000

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)

Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are " well-distributed " in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equi-distribution, as have Fourier analysts when working with the " uncertainty principle ". In this article we find… (More)