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.     

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.     

New inequalities for the Hurwitz zeta function

We establish various new inequalities for the Hurwitz zeta function. Our results generalize some known results for the polygamma functions to the Hurwitz zeta function.     

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.     

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.

     

Riemann zeta function and Lyapunov-type inequalities for certain higher order differential equations

This note generalizes the well known Lyapunov-type inequalities for second-order linear differential equations to certain 2M-th order linear differential equations with five types of boundary conditions. The usage of the best constant of some Sobolev-type inequalities clarify the process for obtaining such inequality and sharpen the result of Çakmak [2].     

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.     

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.

     

A half-discrete Hilbert-type inequality in the whole plane related to the Riemann zeta function

By the use of Hermite–Hadamard’s inequality and weight functions, a half-discrete Hilbert-type inequality in the whole plane with the kernel of hyperbolic cotangent function and multi-parameters is given. The constant factor related to the Riemann zeta function is proved to be the best possible. The equivalent forms, two kinds of particular inequalities, the operator expressions and some equivalent reverses are considered.     

Fourier series for zeta function via Sinc

In this paper we derive some Fourier series and Fourier polynomial approximations to a function F which has the same zeros as the zeta function, ζ(z) on the strip {z∈C:0<Rz<1}. These approximations depend on an arbritrary positive parameter h, and which for arbitrary ε∈(0,1/2), converge uniformly to ζ(z) on the rectangle {z∈C:ε<Rz<1-ε,-π/h<Iz<π/h}.     

A characterization of the zero-free region of the Riemann zeta function and its applications

We characterize the zero-free regions of a class of functions (including the Riemann zeta function) in half-planes in terms of closures of ranges of the corresponding multiplication operators on Hardy spaces. We give an explicit characterization of these closures. As applications, we obtain a weaker version of the Nyman–Beurling–Báez-Duarte criterion, and provide some investigations on a problem relating to the Riemann hypothesis proposed by Báez-Duarte et al. [Adv. Math. 149 (2000) 130-144].     

On the occurrence of the sine kernel in connection with the shifted moments of the Riemann zeta function

We point out an interesting occurrence of the sine kernel in connection with the shifted moments of the Riemann zeta function along the critical line. We discuss rigorous results in this direction for the shifted second moment and for the shifted fourth moment. Furthermore, we conjecture that the sine kernel also occurs in connection with the higher (even) shifted moments and show that this conjecture is closely related to a recent conjecture by Conrey, Farmer, Keating, Rubinstein, and Snaith (2003, 2005) [CFKRS1] and [CFKRS2].     

Zeros of Dedekind zeta functions in the critical strip

In this paper, we describe a computation which established the GRH to height (resp. ) for cubic number fields (resp. quartic number fields) with small discriminant. We use a method due to E. Friedman for computing values of Dedekind zeta functions, we take care of accumulated roundoff error to obtain results which are mathematically rigorous, and we generalize Turing's criterion to prove that there is no zero off the critical line. We finally give results concerning the GRH for cubic and quartic fields, tables of low zeros for number fields of degree and , and statistics about the smallest zero of a number field.

     

Curve Fitting, Differential Equations And The Riemann Hypothesis

It is known that the Riemann zeta function ζ (s) in the critical strip 0 < Re(s) < 1, may be represented as the Mellin transform of a certain function φ (x) which is related to one of the theta functions. The function φ (x) satisfies a well known functional equation, and guided by this property we deduce a family of approximating functions involving an arbitrary parameter α. The approximating function corresponding to the value of α = 2 gives rise to a particularly accurate numerical approximation to the function φ (x). Another approximation to φ (x), which is based upon the first one, is obtained by solving a certain differential equation. Yet another approximating function may be determined as a simple extension of the first. All three approximations, when used in conjunction with the Mellin transform expression for ζ (s) in the critical strip, give rise to an explicit expression from which it is clear that Re(s) = 1/2 is a necessary and sufficient condition for the vanishing of the imaginary part of the integral, the real part of which is non-zero. Accordingly, the analogy with the Riemann hypothesis is only partial, but nevertheless Re(s) = 1/2 emerges from the analysis in a fairly explicit manner. While it is generally known that the imaginary part of the Mellin transform must vanish along Re(s) = 1/2, the major contribution of this paper is the presentation of the actual calculation for three functions which approximate φ (x). The explicit nature of these calculation details may facilitate progress towards the corresponding calculation for the actual φ (x), which may be necessary in a resolution of the Riemann hypothesis.2000 Mathematics Subject Classification: Primary—11M06, 11M26     

Some Unusual Identities for Special Values of the Riemann Zeta Function

In this paper, we use elementary methods to derive some new identities for special values of the Riemann zeta function.