Riemann zeta函数的收敛区域

给出了Riemann zeta函数收敛区域的几种证明.     

Arithmetic progressions of zeros of the Riemann zeta function

If the Riemann zeta function vanishes at each point of the finite arithmetic progression {D+inp}_0<|n|<N (D?1/2, p>0), then N<13p if D=1/2, and N<p^1/D-1+o(1) in general.     

Large gaps between the zeros of the Riemann zeta function

We show that the generalized Riemann hypothesis implies that there are infinitely many consecutive zeros of the zeta function whose spacing is three times larger than the average spacing. This is deduced from the calculation of the second moment of the Riemann zeta function multiplied by a Dirichlet polynomial averaged over the zeros of the zeta function.     

The holomorphic flow of the Riemann zeta function

The flow of the Riemann zeta function, , is considered, and phase portraits are presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest, using the Chebyshev method and using Mathematica.

The phase diagrams suggest new analytic properties of zeta, of which some are proved and others are given in the form of conjectures.

     

On the zeros of the Riemann zeta function of large multiplicity

We obtain a new upper bound for the number of zeros of the Riemann zeta function of a given multiplicity lying in a given rectangle of the critical strip.     

On the difference between consecutive zeros of the Riemann zeta function

Let 0 < γ₁ ≤ γ₂ ≤ … be the imaginary part of the zeros, $\text{λ =}\text{lim}{}_{\mathrm{n}}\text{(γ}{}_{\mathrm{n}}\text{? γ}{}_{\mathrm{n\; ?\; 1}}\text{)(}\text{log}\text{γ}{}_{\mathrm{n}}\text{2π}\text{)}$ and $\text{μ =}\text{lim}{}_{\mathrm{n}}\text{(γ}{}_{\mathrm{n}}\text{? γ}{}_{\mathrm{n\; ?\; 1}}\text{)(}\text{log}\text{γ}{}_{\mathrm{n}}\text{2π}\text{)}$. Assuming the Riemann hypothesis, it is known that μ ≤ 0.68 and λ > 1. One suspects that μ = 0 and λ = +∞. The object of this note is to show that λ ≥ 1.9.     

On some new properties of the gamma function and the Riemann zeta function

In this paper, we have exhibited, by utilizing value distribution theory, some new properties of the Gamma function Γ(z) and the Riemann zeta function ζ(z). Specifically, we have proved that both of the two functions are prime and the Riemann zeta function, like Γ(z), does not satisfy any algebraic differential equation with coefficients in ??₀. Moreover, the two functions do not satisfy any functional equation of the form P(Γ, ζ, z) ≡ 0, where P(x, y, z) is a nonconstant polynomial in x, y and z.     

Short series over simple zeros of the Riemann zeta-function

Recently, Garaev showed that the series Σ_?∥?ζ¹(?)∥⁻¹ diverges, where the sum is taken over the simple zeros ? = β + iγ of the Riemann zeta-function ζ(s). More precisely, he proved . Using a mean-value estimate due to Ramachandra and some result on the distribution of simple zeros in short intervals on the critical line, we prove for T^0.552 ≤ H ≤ T. This leads to a slight improvement of Garaev's result in replacing his lower bound by .     

The two-variable zeta function and the Riemann hypothesis for function fields

We present Bombieri's proof of the Riemann hypothesis for the zeta function of a curve over a finite field. We first briefly describe this zeta function and discuss the two-variable zeta function of Pellikaan. Then we give Naumann's proof that the numerator of this function is irreducible.     

Trace formula on the p-adic upper half-plane

This article aims at showing a p-adic analogue of Selberg's trace formula, which describes a duality between the spectrum of a Hilbert-Schmidt operator and the length of prime geodesics appearing in the p-adic upper half-plane associated with a hyperbolic discontinuous subgroup of . Then we construct Markov processes on the fundamental domain relative to such subgroups, to whose transition operators the trace formula applied and a p-adic analogue of prime geodesic theorem is proved.     

Linear law for the logarithms of the Riemann periods at simple critical zeta zeros

Each simple zero of the Riemann zeta function on the critical line with is a center for the flow of the Riemann xi function with an associated period . It is shown that, as ,

Numerical evaluation leads to the conjecture that this inequality can be replaced by an equality. Assuming the Riemann Hypothesis and a zeta zero separation conjecture for some exponent , we obtain the upper bound . Assuming a weakened form of a conjecture of Gonek, giving a bound for the reciprocal of the derivative of zeta at each zero, we obtain the expected upper bound for the periods so, conditionally, . Indeed, this linear relationship is equivalent to the given weakened conjecture, which implies the zero separation conjecture, provided the exponent is sufficiently large. The frequencies corresponding to the periods relate to natural eigenvalues for the Hilbert-Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis.

     

Continued-fraction expansions for the Riemann zeta function and polylogarithms

It appears that the only known representations for the Riemann zeta function in terms of continued fractions are those for and 3. Here we give a rapidly converging continued-fraction expansion of for any integer . This is a special case of a more general expansion which we have derived for the polylogarithms of order , , by using the classical Stieltjes technique. Our result is a generalisation of the Lambert-Lagrange continued fraction, since for we arrive at their well-known expansion for . Computation demonstrates rapid convergence. For example, the 11th approximants for all , , give values with an error of less than 10.

     

The joint value-distribution of the Riemann zeta function and Hurwitz zeta functions

In this paper, we investigate the joint value-distribution for the Riemann zeta function and Hurwitz zeta function attached with a transcendental real parameter. Especially, we establish the joint universality theorem for these two zeta functions. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 39–57, January–March, 2007.     

Localization of the first zero of the Dedekind zeta function

Using Weil's explicit formula, we propose a method to compute low zeros of the Dedekind zeta function. As an application of this method, we compute the first zero of the Dedekind zeta function associated to totally complex fields of degree less than or equal to 30 having the smallest known discriminant.

     

A generalized Ihara zeta function formula for simple graphs with bounded degree

We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is a generalization of the formula obtained by G. Chinta, J. Jorgenson and A. Karlsson from vertex-transitive graphs.     

Remark on a double-inequality for the Riemann zeta function

Let ζ be the Riemann zeta function and δ(x)=1/(2^x-1). For all x>0 we have

(1-δ(x))ζ(x)+αδ(x)<ζ(x+1)<(1-δ(x))ζ(x)+βδ(x),