ErgodicPNT

We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action on the torsion points of an abelian variety. The resulting upper bounds require explicit character sum calculations, with stronger results holding if one assumes the Generalized Riemann Hypothesis.

Résumé - AbstractLet K=ℚ(-D) be an imaginary quadratic field, and denote by h its class number. It is shown that there is an absolute constant c>0 such that for sufficiently large D at least c·h∏ p∣D (1-p -1 ) of the h distinct L-functions L K (s,χ) do not vanish at the central point s=1/2.

Are there infinitely many prime pairs with given even difference? Most mathematicians think so. Using a strong arithmetic hypothesis, Goldston, Pintz and Yildirim have recently shown that there are infinitely many pairs of primes differing by at most sixteen.There is extensive numerical support for the prime-pair conjecture (PPC) of Hardy and Littlewood [G.H. Hardy, J.E. Littlewood, Some problems of ‘partitio numerorum’. III: On the expression of a number as a sum of primes, Acta Math. 44 (1923) 1–70 (sec. 3)] on the asymptotic behavior of π2r(x), the number of prime pairs with p≤x. Assuming Riemann’s Hypothesis (RH), Montgomery and others have studied the pair-correlation of zeta’s complex zeros, indicating connections with the PPC. Using a Tauberian approach, the author shows that the PPC is equivalent to specific boundary behavior of a function involving zeta’s complex zeros. A certain hypothesis on equidistribution of prime pairs, or a speculative supplement to Montgomery’s work on pair-correlation, would imply that there is an abundance of prime pairs.

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We now turn to several specific examples of this principle in various contexts.  We begin with the more “combinatorial” or “non-ergodic theoretical” instances of this principle, in which there is no underlying probability measure involved; these situations are simpler than the ergodic-theoretic ones, but already illustrate many of the key features of this principle in action.

An alternate argument for the arithmetic large sieve inequality September 2008 This short note describes a very natural and well-motivated derivation of the "arithmetic" large sieve inequality from the dual of the analytic inequality, which avoids the usual trick of submultiplicativity of Gallagher. This is also described in a blog post.

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