2006 Fiscal Year Final Research Report Summary
The Riemann zeta-function : its embedding into the Hilbert space over a Lie group
| Project/Area Number |
15540047
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Nihon University |
|---|
Principal Investigator |
MOTOHASHI Yoichi Nihon University, College of Science and Technology, Professor, 理工学部, 教授 (30059969)
|
|---|
| Project Period (FY) |
2003 – 2006
|
| Keywords | automorphic representation / zeta-function / distribution of primes |
|---|
| Research Abstract |
The achievements are divided into two categories : the theory of L-functions and the theory of the distribution of prime numbers. The principal result is the proof of the fundamental assertion that the Lindeloef constant for any automorphic L-function is less than or equal to 1/3. The details was published, as a joint work with M.Jutila, in Acta Mathematica of the Royal Swedish Academy (vol.195 (2005)), one of the most internationally acclaimed periodicals in pure mathematics. An extension to any Rankin-Selberg L-function was soon made, again jointly with Jutila ; this time, the constant is less than or equal to 2/3, which, despite a non-uniformity to an extent, supersedes considerably the hitherto best result due to the school of number theorists at Princeton. The work was published orally by Jutila at the international conference 'Multiple Zeta-Functions' (2005) ; Motohashi was unable to participate because of his teaching duty. The details was published together with Motohashi's exp
… More
ository article on the mean values of zeta-functions, in a famed proceedings volume of the American Mathematical Society (2006). The achievement of those constants or exponents is now appreciated by most specialists as to be very hard to go beyond. Recently, in the theory of the distribution of prime numbers were made extraordinary discoveries. One of them is due to D.A.Goldston, J.Pintz, and C.Y.Yildirim. They found a highly promising way to come close to the solution of the famous 'Twin Prime Conjecture' by showing the existence of small gaps between prime numbers. Motohashi made a contribution by devising an exceptionally short proof of the core part of their result, and the details were published as a quadruple-authored paper in the Proceedings of Japan Academy (2006). Later, Motohashi and Pintz obtained an ideal improvement of the relevant sieve method ; the result is to be published at an international conference to be held at Columbia University (N.Y.), provided Motohashi's teaching schedule allows him to attend the meeting. Albeit this is after the closing of the research, very recently the 14^<th> article (2004) of Motohashi's widely known series on mean values of the zeta and L-functions was exploited in a fundamental way by V.Blomer and G.Harcos. Being inspired by this advance, Motohashi has devised an alternative proof of their result ; and based on it, he has established a complete spectral decomposition of the mean square of the automorphic L-function attached to any irreducible representation in the unitary principal series-in a unified fashion. This is a solution to a long-standing problem concerning L-functions. Motohashi envisages an emergence of a grand theory on such mean values and its applications to the distribution of prime numbers. Less
|
