希尔伯特－波利亚猜想

历史

$\tfrac12 + it$

1970年代与随机矩阵

1970年代初，蒙哥马利发现了临界线上非平凡零点统计分布的规律，被称为蒙哥马利对关联假设（Montgomery's pair correlation conjecture）。他发现非平凡零点之间并不靠近，而是有互相排斥的趋势。1972年，在他访问普林斯顿高等研究院时，他将其成果告诉了随机矩阵专家弗里曼·戴森

与量子力学的可能联系

$E_{n}=E_{n}^{0}+ \langle \phi^{0}_n \vert V \vert \varphi^{0}_n \rangle$

$V(x)=A\int_{-\infty}^{\infty} (g(k)+\overline{g(k)}-E_{k}^{0})\,R(x,k)\,dk$

$g(k)=i \sum_{n=0}^{\infty} \left(\frac{1}{2}-\rho_n \right)\delta(k-n)$

$H = \tfrac1{2} (xp+px) = - i \left( x \frac{\mathrm{d}}{\mathrm{d} x} + \frac1{2} \right).$

参考文献

• Aneva B., "Symmetry of the Riemann operator", (1999) Physics Letters, B450: 388–396.
• Berry, Michael V.; Keating, Jonathan P., H = xp and the Riemann zeros, (编) Keating, Jonathan P.; Khmelnitski, David E.; Lerner, Igor V., Supersymmetry and Trace Formulae: Chaos and Disorder, New York: Plenum, 355–367, 1999a年, ISBN 978-0-306-45933-7 .
• Montgomery, Hugh L., The pair correlation of zeros of the zeta function, Analytic number theory, Proc. Sympos. Pure Math., Providence, R.I.: American Mathematical Society, 181–193, 1973年, MR0337821
• Berry, M.V.; Keating, J.P. (1999b), "The Riemann zeros and eigenvalue asymptotics", SIAM Review, 41(2): 236–266.
• Zeev Rudnick; Peter Sarnak (1996), "Zeros of Principal L-functions and Random Matrix Theory", Duke Journal of Mathematics, 81: 269–322.
• Elizalde Emilio ; 'Zeta regularization techniques with applications' ISBN 978-981-02-1441-8981-02-1441-3, here the author explain in what sense the problem of HIlbert-Polya is related with the problem of Gutzwiller Trace formula and what would be the value of the sum $\exp(i\gamma)$ taken over the imaginary parts of the zeros.