scholarly journals Dirichlet Series with Periodic Coefficients and Their Value-Distribution near the Critical Line

2021 ◽  
Vol 314 (1) ◽  
pp. 238-263
Author(s):  
Athanasios Sourmelidis ◽  
Jörn Steuding ◽  
Ade Irma Suriajaya
Keyword(s):  
Dirichlet Series ◽  
Critical Line ◽  
Value Distribution ◽  
Periodic Coefficients

Dirichlet series with periodic coefficients

1999 ◽  
Vol 35 (1-2) ◽  
pp. 70-88 ◽  
Author(s):  
Makoto Ishibashi ◽  
Shigeru Kanemitsu
Keyword(s):  
Dirichlet Series ◽  
Periodic Coefficients

On the Value Distribution of Shifts of Universal Dirichlet Series

2006 ◽  
Vol 147 (4) ◽  
pp. 309-317 ◽  
Author(s):  
Jerzy Kaczorowski ◽  
Antanas Laurinčikas ◽  
Jörn Steuding
Keyword(s):  
Dirichlet Series ◽  
Value Distribution

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

1995 ◽  
Vol 450 (1939) ◽  
pp. 439-462 ◽  
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 .

Value distribution of general Dirichlet series. VII

2006 ◽  
Vol 46 (2) ◽  
pp. 155-162 ◽  
Author(s):  
J. Genys ◽  
A. Laurinčikas ◽  
D. Šiaučiūnas
Keyword(s):  
Dirichlet Series ◽  
Value Distribution ◽  
General Dirichlet Series

Chowla’s theorem over function fields

2018 ◽  
Vol 14 (06) ◽  
pp. 1689-1698
Author(s):  
Yoshinori Hamahata
Keyword(s):  
Dirichlet Series ◽  
Function Field ◽  
Function Fields ◽  
Rational Numbers ◽  
Periodic Coefficients ◽  
Linearly Independent ◽  
Parity Condition

Sarvadaman Chowla proved that if [Formula: see text] is an odd prime, then [Formula: see text] ([Formula: see text]) are linearly independent over the field of rational numbers. We establish an analog of this result over function fields. As an application, we prove an analog of the Baker–Birch–Wirsing theorem about the non-vanishing of Dirichlet series with periodic coefficients at [Formula: see text] in the function field setup with a parity condition.

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

1992 ◽  
Vol 437 (1899) ◽  
pp. 151-173 ◽  
Keyword(s):  
Zeta Function ◽  
Analytic Continuation ◽  
Dirichlet Series ◽  
Asymptotic Representation ◽  
Critical Line ◽  
Error Function ◽  
Real Variable ◽  
Computational Effort ◽  
Selberg Zeta Function ◽  
Semiclassical Approximations

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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