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A new asymptotic representation for $\zeta \left(\frac{1}{2}+it\right)$ and quantum spectral determinants. (English) Zbl 0776.11048

The authors describe a new way, which has some advantages over the methods currently used, in order to calculate with great accuracy the zeta function $\zeta \left(1/2+it\right)$ in the critical strip $|\text{Im}\phantom{\rule{0.166667em}{0ex}}t|<1/2$, $|\text{Re}\phantom{\rule{0.166667em}{0ex}}t|\gg 1$. From there, the critical line, $t$ real, can be approached (this is where, according to the Riemann hypothesis, the non-trivial lie). The method used is an extension of the formal approximation scheme introduced by the second author [Proc. R. Soc. Lond. A 436, 99–108 (1992; Zbl 0820.58057)]. By analytical continuation of the Dirichlet series for $\zeta \left(s\right)$ to the critical line $s=1/2+it$, $t$ real, the authors find a family of exact representations, parametrized by a real variable $K$, for the real function $Z\left(t\right)=\zeta \left(1/2+it\right)exp\left[i\theta \left(t\right)\right]$, with $\theta$ real. Computation of $Z\left(t\right)$ for real $t$ starting from the ordinary series representation for the Riemann zeta function $\zeta \left(s\right)$ requires in fact, analytic continuation, which can be done by using the Riemann-Siegel formula. The dominant contribution obtained by the authors, ${Z}_{0}\left(t,K\right)$, can be expressed as a convergent sum over the integers $n$ of the Dirichlet series, resembling the finite main sum of the Riemann-Siegel formula but with the sharp cut-off smoothed by an error function. The corrections ${Z}_{3}\left(t,K\right)$, ${Z}_{4}\left(t,K\right),\cdots$ are also convergent sums, whose principal terms involve integers close to the Riemann-Siegel cut-off, and for large $K$, ${Z}_{0}$ contains not only the main sum of the Riemann-Siegel formula but also its first correction.

Graphical and numerical investigations show explicitly that the ${Z}_{0}$ obtained by the authors is always better than the main sum of the Riemann-Siegel formula, and this is obtained with little more computational effort. By regarding Planck’s constant $\hslash$ as a complex variable, the method of the authors for $Z$ can be directly applied to obtain semiclassical approximations for spectral determinants whose zeros are the energies of stationary states in quantum mechanical problems, such as the ones involved in the quantization of chaos. In this way, they can analytically continue the divergent Gutzwiller trace formula and obtain, in principle, an exact asymptotic representation of the Selberg zeta function on its critical line.

##### MSC:
 11M26 Nonreal zeros of $\zeta \left(s\right)$ and $L\left(s,\chi \right)$; Riemann and other hypotheses 81Q50 Quantum chaos 11M06 $\zeta \left(s\right)$ and $L\left(s,\chi \right)$ 11M36 Selberg zeta functions and regularized determinants