A refinement of theorems on the number of zeros lying on intervals of the critical line of certain Dirichlet series

1992 ◽  
Vol 47 (2) ◽  
pp. 219-220 ◽  
Author(s):  
A A Karatsuba
Keyword(s):  
Dirichlet Series ◽  
Critical Line ◽  
Number Of Zeros

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 .

scholarly journals On the zeros of a class of generalised Dirichlet series-VIII

1991 ◽  
Vol Volume 14 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra
Keyword(s):  
Lower Bound ◽  
Dirichlet Series ◽  
Number Of Zeros ◽  
International Audience

International audience In an earlier paper (Part VII, with the same title as the present paper) we proved results on the lower bound for the number of zeros of generalised Dirichlet series $F(s)= \sum_{n=1}^{\infty} a_n\lambda^{-s}_n$ in regions of the type $\sigma\geq\frac{1}{2}-c/\log\log T$. In the present paper, the assumptions on the function $F(s)$ are more restrictive but the conclusions about the zeros are stronger in two respects: the lower bound for $\sigma$ can be taken closer to $\frac{1}{2}-C(\log\log T)^{\frac{3}{2}}(\log T)^{-\frac{1}{2}}$ and the lower bound for the number of zeros is something like $T/\log\log T$ instead of the earlier bound $>\!\!\!>T^{1-\varepsilon}$.

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.

scholarly journals AN ELECTROSTATIC DEPICTION OF THE VALIDITY OF THE RIEMANN HYPOTHESIS AND A FORMULA FOR THE NTH ZERO AT LARGE N

2013 ◽  
Vol 28 (30) ◽  
pp. 1350151 ◽  
Author(s):  
ANDRÉ LECLAIR
Keyword(s):  
Electric Field ◽  
Critical Line ◽  
Quantum Statistical Mechanics ◽  
Spatial Dimension ◽  
Transcendental Equation ◽  
Critical Strip ◽  
Number Of Zeros ◽  
Large N ◽  
Relativistic Gases ◽  
Counting Formula

We construct a vector field E from the real and imaginary parts of an entire function ξ(z) which arises in the quantum statistical mechanics of relativistic gases when the spatial dimension d is analytically continued into the complex z plane. This function is built from the Γ and Riemann ζ functions and is known to satisfy the functional identity ξ(z) = ξ(1-z). E satisfies the conditions for a static electric field. The structure of E in the critical strip is determined by its behavior near the Riemann zeros on the critical line ℜ(z) = 1/2, where each zero can be assigned a ⊕ or ⊖ parity, or vorticity, of a related pseudo-magnetic field. Using these properties, we show that a hypothetical Riemann zero that is off the critical line leads to a frustration of this "electric" field. We formulate this frustration more precisely in terms of the potential Φ satisfying E = -∇Φ and construct Φ explicitly. The main outcome of our analysis is a formula for the nth zero on the critical line for large n expressed as the solution of a simple transcendental equation. Riemann's counting formula for the number of zeros on the entire critical strip can be derived from this formula. Our result is much stronger than Riemann's counting formula, since it provides an estimate of the nth zero along the critical line. This provides a simple way to estimate very high zeros to very good accuracy, and we estimate the 10106-th one.

scholarly journals Zeros of the Epstein zeta function to the right of the critical line

2020 ◽  
pp. 1-12
Author(s):  
YOUNESS LAMZOURI
Keyword(s):  
Zeta Function ◽  
Error Term ◽  
Critical Line ◽  
Quadratic Field ◽  
Positive Definite ◽  
Real Numbers ◽  
Positive Definite Quadratic Form ◽  
Number Of Zeros ◽  
Epstein Zeta Function ◽  
The Right

Abstract Let E(s, Q) be the Epstein zeta function attached to a positive definite quadratic form of discriminant D < 0, such that h(D) ≥ 2, where h(D) is the class number of the imaginary quadratic field ${{\mathbb{Q}}(\sqrt D)}$ . We denote by N E (σ1, σ2, T) the number of zeros of E(s, Q) in the rectangle σ1 < Re(s) ≤ σ2 and T ≤ Im (s) ≤ 2T, where 1/2 < σ1 < σ2 < 1 are fixed real numbers. In this paper, we improve the asymptotic formula of Gonek and Lee for N E (σ1, σ2, T), obtaining a saving of a power of log T in the error term.

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