A new asymptotic representation for ζ(½ + i
            t
            ) and quantum spectral determinants

By analytic continuation of the Dirichlet series for the Riemann zeta function ζ(s) to the critical line s = ½ + i t ( t real), a family of exact representations, parametrized by a real variable K , is found for the real function Z ( t ) = ζ(½ + i t ) exp {iθ( t )}, where θ 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 ≫ 1 when K < t 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 { ─½ 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 h as a complex variable, the method for Z ( t ) can be applied directly to semiclassical approximations for spectral determinants ∆( E, h ) whose zeros E = E j ( h ) 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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An asymptotic representation for the Riemann zeta function on the critical line

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1994.0121 ◽

1994 ◽

Vol 446 (1928) ◽

pp. 565-587 ◽

Cited By ~ 7

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Asymptotic Representation ◽

Critical Line ◽

Error Function ◽

Computational Effort ◽

Incomplete Gamma Functions ◽

The Riemann Zeta Function ◽

Convergent Expansion ◽

Riemann Zeta

A representation for the Riemann zeta function ζ( s ) is given as an absolutely convergent expansion involving incomplete gamma functions which is valid for all finite complex values of s (≠ 1). It is then shown how use of the uniform asymptotics of the incomplete gamma function leads to a new asymptotic representation for ζ( s ) on the critical line s = ½ + i t when t → ∞. This new result involves an error function smoothing of an infinite sum and consequently shares some similarity to, though is quite different from, the recent asymptotic expansion for ζ(½ + i t ) developed by Berry & Keating. Numerical examples suggest that term for term (with a little extra computational effort) the new representation is at least as accurate as the Riemann–Siegel formula.

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From classical periodic orbits to the quantization of chaos

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1992.0087 ◽

1992 ◽

Vol 437 (1901) ◽

pp. 693-714 ◽

Cited By ~ 11

Keyword(s):

Periodic Orbit ◽

Zeta Function ◽

Dirichlet Series ◽

Periodic Orbits ◽

Trace Formula ◽

Riemann Surfaces ◽

Chaotic Systems ◽

Critical Line ◽

Selberg Zeta Function ◽

Periodic Orbit Theory

We present a quantitative analysis of Selberg’s trace formula viewed as an exact version of Gutzwiller’s semiclassical periodic-orbit theory for the quantization of classically chaotic systems. Two main applications of the trace formula are discussed in detail, (i) The periodic-orbit sum rules giving a smoothing of the quantal energy-level density. (ii) The Selberg zeta function as a prototype of a dynamical zeta function defined as an Euler product over the classical periodic orbits and its analytic continuation across the entropy barrier by means of a Dirichlet series. It is shown how the long periodic orbits can be effectively taken into account by a universal remainder term which is explicitly given as an integral over an ‘orbit-selection function’. Numerical results are presented for the free motion of a point particle on compact Riemann surfaces (Hadamard-Gutzwiller model), which is the primary testing ground for our ideas relating quantum mechanics and classical mechanics in the case of strong chaos. Our results demonstrate clearly the crucial role played by the long periodic orbits. An exact rule for quantizing chaos is derived for such systems where the Dirichlet series representing the Selberg zeta function converges on the critical line. Explicit formulae are given for the computation of the abscissae of absolute and conditional convergence, respectively, of these dynamical Dirichlet series. For the two Riemann surfaces considered, it turns out that one can cross the entropy barrier, but that the critical line cannot be reached by a convergent Dirichlet series. It would seem that this is the main reason why the Riemann-Siegel lookalike formula, recently conjectured by M. V. Berry and J. P. Keating, fails in generating the lower-lying quantal energies for these strongly chaotic systems.

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The Riemann-Siegel expansion for the zeta function: high orders and remainders

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1995.0093 ◽

1995 ◽

Vol 450 (1939) ◽

pp. 439-462 ◽

Cited By ~ 8

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Critical Line ◽

High Accuracy ◽

Stokes Line ◽

Power Dependence ◽

Multiplier Function ◽

Riemann’S Zeta Function

On the critical line s ═ ½ + i t ( t real), Riemann’s zeta function can be calculated with high accuracy by the Riemann-Siegel expansion. This is derived here by elementary formal manipulations of the Dirichlet series. It is shown that the expansion is divergent, with the high orders r having the familiar 'factorial' divided by power' dependence, decorated with an unfamiliar slowly varying multiplier function which is calculated explicitly. Terms of the series decrease until r ═ r * ≈ 2π t and then increase. The form of the remainder when the expansion is truncated near r * is determined; it is of order exp(-π t ), indicating that the critical line is a Stokes line for the Riemann-Siegel expansion. These conclusions are supported by computations of the first 50 coefficients in the expansion, and of the remainders as a function of truncation for several values of t .

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Zeta-regularization of arithmetic sequences

EPJ Web of Conferences ◽

10.1051/epjconf/202024401008 ◽

2020 ◽

Vol 244 ◽

pp. 01008

Author(s):

Jean-Paul Allouche

Keyword(s):

Zeta Function ◽

Analytic Continuation ◽

Dirichlet Series ◽

Infinite Product ◽

The Other ◽

Arithmetic Functions ◽

Finite Product ◽

Arithmetic Sequences ◽

Classical Arithmetic

Is it possible to give a reasonable value to the infinite product 1 × 2 × 3 × · · · × n × · · · ? In other words, can we define some sort of convergence of the finite product 1 × 2 × 3 × · · · × n when n goes to infinity? One way is to relate this product to the R√iemann zeta function and to its analytic continuation. This approach leads to: 1 × 2 × 3 × · · · × n × · · · = 2π. More generally the “zeta-regularization” of an infinite product consists of introducing a related Dirichlet series and its analytic continuation at 0 (if it exists). We will survey some properties of this generalized product and allude to applications. Then we will give two families of possibly new examples: one unifies and generalizes known results for the zeta-regularization of the products of Fibonacci, balanced and Lucas-balanced numbers; the other studies the zeta-regularized products of values of classical arithmetic functions. Finally we ask for a possible zeta-regularity notion of complexity.

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ON THE RANKIN–SELBERG ZETA FUNCTION

Journal of the Australian Mathematical Society ◽

10.1017/s1446788712000225 ◽

2012 ◽

Vol 93 (1-2) ◽

pp. 101-113

Author(s):

ALEKSANDAR IVIĆ

Keyword(s):

Functional Equation ◽

Zeta Function ◽

Critical Line ◽

Critical Strip ◽

Approximate Functional Equation ◽

Selberg Zeta Function

AbstractWe obtain the approximate functional equation for the Rankin–Selberg zeta function in the critical strip and, in particular, on the critical line $\operatorname {Re} s= \frac {1}{2}$.

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Alternative numerical systems

Journal of Nature, Science & Technology ◽

10.36937/janset.2021.003.002 ◽

2021 ◽

Vol 1 (3) ◽

pp. 6-10

Author(s):

Yuriy N. Zayko

Keyword(s):

Dark Energy ◽

Zeta Function ◽

Analytic Continuation ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

Real Numbers ◽

Complex Argument ◽

System A ◽

The Riemann Zeta Function ◽

Riemann Zeta

The article is devoted to the construction of numerical systems, alternative to the system of real numbers and applicable in curvilinear space-time. Examples of such systems are given. Within the framework of a stationary numerical system, it is admissible to sum the diverging series like the Dirichlet series for the Riemann zeta function without resorting to its analytic continuation in the plane of the complex argument. In the framework of a non-stationary numerical system, a description of the Hubble effect is obtained, taking into account the corrections that correspond to the apparently accelerated recession of galaxies without invoking the hypothesis of dark energy.

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