Unnormalized differences between zeros of L-functions. (24th October 2014)

• Record Type: Journal Article
• Authors: Ford, Kevin; Zaharescu, Alexandru
• Is Part Of: Compositio mathematica. Volume 151:Number 2(2015:Mar.)
• Page Start: 230
• Page End: 252
• Publication Date: 2014-10-24
• Subjects: 11M26 (primary), -- 11K38 (secondary); Riemann zeta function, -- L-functions, -- distribution of zeros; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X14007659
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract: We study a subtle inequity in the distribution of unnormalized differences between imaginary parts of zeros of the Riemann zeta function, which was observed by a number of authors. We establish a precise measure which explains the phenomenon, that the location of each Riemann zero is encoded in the distribution of large Riemann zeros. We also extend these results to zeros of more general $L$ -functions. In particular, we show how the rank of an elliptic curve over $\mathbb{Q}$ is encoded in the sequences of zeros of other $L$ -functions, not only the one associated to the curve.