Japanese version
Mr. Woit's blog was introduced ignited by Dr. Witten's lecture at IAS for general: "Knots and Quantum Theory."
About physics and arithmetic, Dr. Witten might say that there would be a bridge between Number Theory and Physics.
Of course, the keyword is Langlands. The magazine "arithmetic and physics" (I'm forgotten the formal name of English journal) is started, and the first publication and the leadoff article are Witten-Kapustin. Indeed!
Dr. Sarnak's lecture note contains Selberg 1/4 conjecture, general Ramanujan anticipation, etc. about which are argued on Woit's Blog.report. The following lecture notes by Dr. Sarnak are not familiar in Japan.
1,"Arinthmetic Quantum Chaos"
2,"Selberg eigenvalue problems"
3,"The generalized Ramanujan conjecture"
4,"Spectra of hyperbolic surfaces"
the above item 4 had not come out in Woit's BLOG but I think it is a very interesting and exiting note and let me it join to others. It is pointing the direction of "physics and arithmetic." Its contents are:
1,Introduction
2,Existence
3,High Energy Spectrum
4,Low Energy Spectrum
In 2004 Dr. Sarnak published a article about Riemann Hypothesis on Claymath Instiute.
Problems of the Millennium: The Riemann Hypothesis (2004)
May/27/2011 added：
In 1993 Dr. Sarnak had proved the analogy of Shimura correspondense for Maass forms (with SVETLANA KATOK）．
HEEGNER POINTS, CYCLES AND MAASS FORMS
Someone taught me that the "Arinthmetic and Physics" journal is the following:
ReplyDeleteCommunications in Number Theory and Physics
Volume 1 (2007) Issue 1 (March) Number 1
"Electric-Magnetic Duality And The Geometric Langlands Program" Anton Kapustin and Edward Witten p.1-236
Thank you very much!
The Original announcement article was written by E. Bombieri.
ReplyDeletehttp://www.claymath.org/millennium/Riemann_Hypothesis/riemann.pdf
Both are very interesting.
The problem of Mr. Sarnak of "the problems of arithmetic, analysis, and mathematical physics."
ReplyDelete1,The zero point of arbitrary L(s,f) has a real part equal to one half (general Riemann Hypothesis).
2,Find the constant c (>0) independent on q so that it saisfies the following.
\[ L(s,\chi_q)\ge\frac{c}{\log{q}} \]
3,The rank of E(\mathbb{Q}) is as equal to a number as that of the zero point in s=1/2 of L(s,f). (BSD conjecture)
4,Let L(s,f) be fixed, the local interval distribution during the zero point in a high position will serve as GUE.
5, Generalized Ramanujan Conjecture
6,Functoriality and a Langlands program
7,Probably, twin primes exist infinitely.
8,For the big integer N are there some algorithm to factorized into prime factor, from which time complexity serves as a polynomial order?
9,What happens to the asymptotic action in n\rightarrow\inftyn\rightarrow\infty of \Delta(n)? In particular, is the following realized?
\[\lim_{n\rightarrow\infty}\frac{\log_2{\Delta}(n)}{n}=-1. \]
As mentioned above, anticipation towards the 21st century of Dr. Sarnak