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A New Asymptotic Representation for $\zeta $($\frac{1}{2}$+it) and Quantum Spectral Determinants

M. V. Berry, J. P. Keating


By analytic continuation of the Dirichlet series for the Riemann zeta function $\zeta $(s) to the critical line s = $\frac{1}{2}$ + it (t real), a family of exact representations, parametrized by a real variable K, is found for the real function Z(t) = $\zeta $($\frac{1}{2}$+it) exp {i$\theta $(t)}, where $\theta $ is real. The dominant contribution Z$_{0}$ (t, K) is a convergent sum over the integers n of the Dirichlet series, resembling the finite `main sum' of the Riemann-Siegel formula (RS) but with the sharp cut-off smoothed by an error function. The corrections Z$_{3}$(t, K), Z$_{4}$(t, K)... are also convergent sums, whose principal terms involve integers close to the RS cut-off. For large K, Z$_{0}$ contains not only the main sum of RS but also its first correction. An estimate of high orders m $\gg $ 1 when K < t$^{\frac{1}{6}}$ shows that the corrections Z$_{k}$ have the `factorial/power' form familiar in divergent asymptotic expansions, the least term being of order exp {-$\frac{1}{2}$K$^{2}$t}. Graphical and numerical exploration of the new representation shows that Z$_{0}$ is always better than the main sum of RS, providing an approximation that in our numerical illustrations is up to seven orders of magnitude more accurate with little more computational effort. The corrections Z$_{3}$ and Z$_{4}$ give further improvements, roughly comparable to adding RS corrections (but starting from the more accurate Z$_{0}$). The accuracy increases with K, as do the numbers of terms in the sums for each of the Z$_{m}$. By regarding Planck's constant $\hslash $ as a complex variable, the method for Z(t) can be applied directly to semiclassical approximations for spectral determinants $\Delta $(E, $\hslash $) whose zeros E = E$_{j}$($\hslash $) are the energies of stationary states in quantum mechanics. The result is an exact analytic continuation of the exponential of the semiclassical sum over periodic orbits given by the divergent Gutzwiller trace formula. A consequence is that our result yields an exact asymptotic representation of the Selberg zeta function on its critical line.

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