On the Hurwitz Zeta Functions with Algebraic Irrational Parameter

The property of zeta functions on mixed joint universality in the Voronin’s sense states that any two holomorphic functions can be approximated simultaneously with an accuracy of ε>0 by suitable vertical shifts of the pair consisting the Riemann and Hurwitz zeta functions. A rather general result can be obtained for the classes of zeta functions, particularly when an approximating pair is composed of the Matsumoto zeta functions’ class and the periodic Hurwitz zeta function. In this paper, we prove that this set of shifts has a strict positive density for all but at most countably ε>0. Moreover, we provide concluding remarks on certain more general mixed tuples of zeta functions.

In this paper, we study analytical and arithmetical properties of the twisted zeta function \(\Gamma (s)^{ - 1} \int_0^\infty {e^{ - xt} t^{s - 1} } \prod\nolimits_{j = 1}^N {\frac{{a_j t - \log (w^a j)}} {{1 - w^{a_j } e^{a_j t} }}dt} \), where ℜ(s) > N, ℜ(x) > 0, w ∈ ℂ\{0}, N ∈ ℤ, and a 1, …, a N have positive real parts. These functions have many interesting properties. We prove a collection of fundamental identities satisfied by zeta functions of this kind. For instance, special values of these zeta functions are related to twisted Barnes numbers and polynomials. This gives us a new elementary approach to new and known results concerning the Barnes zeta functions. In particular, we derive some well-known results on the Hurwitz zeta functions.

In this chapter some explicit applications to the regularization, by means of Hurwitz zeta-functions considered in previous chapters, of different problems which have appeared recently in the physical literature, are considered. This kind of zeta functions show up profusely in many applications of quantum physics where regularization techniques are needed; in particular, when one deals with a massive quantum field theory in a (totally or partially) compactified spacetime (spherical or toroidal compactification, for instance). Aside from the interest that a detailed mathematical study of these functions may have on its own (e.g., in number theory), what is actually needed for most physical applications is always the numerical value of these functions, and of their derivatives with respect to the main variable and some of the accompanying parameters, before proceeding with the analytical continuation of the functions and also with the treatment of the (possible) poles. This procedure has already been checked in several situations, where the result is usually given in terms of sums of Hurwitz zeta functions and, generically, under the form of asymptotic expansions. Actually, this situation has been described in some detail in Chap. 4 already. In the present one, five additional, completely different applications are dealt with. Namely, in the first section, the calculation of the Casimir energy corresponding to compact universes without boundary. In the second, the calculation of the sum over one-loop integrals which yields the cross section of a scattering process in a Kaluza–Klein model with spherical compactification. Another application is the study of the critical behavior of a field theory at non-zero temperature. As the fourth example of this chapter, the quantization of two-dimensional gravity by means of the Wheeler–De Witt equation is discussed. Finally, the last case considered is the use of the spectral zeta function for both scalar and vector fields on a spacetime with a noncommutative toroidal part.

On some series representations of the Hurwitz zeta function

There is a well-known formula for the Hurwitz zeta function which implies the functional equation for the Riemann zeta function. We give a new proof of that formula and recover certain integral representations for the Hurwitz and Riemann zeta functions.

In the present paper, we prove some equivalent statements of a Hilbert-type integral inequality in the whole plane with intermediate variables. In our theorems, the constant factor is associated to the Hurwitz zeta function and we prove that it is the best possible. We also derive various special cases and applications.

In this work we apply the classical Plana summation formula to derive integral representations for certain infinite series that their general terms containing Gamma, q q -Gamma, theta, Bessel, and Hurwitz zeta functions.

We study the uncertainty principle associated with the Klein–Gordon equation. As in the previous work [Ann. of Math. 173 (2011)], we consider vanishing along a lattice-cross. The following variants appear naturally: (1) vanishing only along “half” of the lattice-cross, where the “half” is defined as being on the boundary of a quarter-plane, and (2) that the function vanishes on the whole lattice-cross, but we require the function to have Fourier transform supported by one of the two branches of the hyperbola. In case (1) the critical phenomenon is whether the given condition forces the function to vanish on the quarter-plane in question. Here it turns out to be crucial whether the quarter-plane is space-like or time-like, and in short the answer is yes for space-like and no for time-like. The analysis brings us quite far, involving the orbit of the Hilbert kernel under the iterates of the transfer operator, and uses methods from the theory of totally positive matrices as well as Hurwitz zeta functions, and is partially postponed to a separate publication. In case (2), the critical phenomenon occurs at another density, and the dynamics then comes from the standard Gauss transformation t \mapsto 1/t \: \mathrm {mod} \: \mathbb Z on the interval [0, 1]. In the intermediate range of the density of the lattice-cross, we obtain unique extendability of the Fourier transform from one branch of the hyperbola to the other.

Let $$\mathbb {R}_+$$ ℝ + denote the open half-line (0, ∞) and let Δ denote the forward difference operator on the space of functions from $$\mathbb {R}_+$$ ℝ + to $$\mathbb {R}$$ ℝ . In this book, we are interested in the classical difference equation Δf = g on $$\mathbb {R}_+$$ ℝ + , which can be written explicitly as $$\displaystyle f(x+1)-f(x) ~=~ g(x),\qquad x>0, $$ f ( x + 1 ) − f ( x ) = g ( x ) , x > 0 , where $$g\colon \mathbb {R}_+\to \mathbb {R}$$ g : ℝ + → ℝ is a given function. This equation appears naturally in the theory of the Euler gamma function, with $$f(x) = \ln \Gamma (x)$$ f ( x ) = ln Γ ( x ) and $$g(x) = \ln x$$ g ( x ) = ln x , but also in the study of many other special functions such as the Barnes G-function and the Hurwitz zeta function (see Examples 1.6 and 1.7 below).

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Nonlocal continuum analysis of a nonlinear uniaxial elastic lattice system under non-uniform axial load

We present several asymptotic analyses for quantities associated with the Riemann and Hurwitz zeta functions. We first determine the leading asymptotic behavior of the Stieltjes constants γ k ( a ) \gamma _k(a) . These constants appear in the regular part of the Laurent expansion of the Hurwitz zeta function. We then use asymptotic results for the Laguerre polynomials L n α L_n^\alpha to investigate a certain sum S γ ( n ) S_\gamma (n) involving the constants γ k ( 1 ) \gamma _k(1) that appears in application of the Li criterion for the Riemann hypothesis. We confirm the sublinear growth of S γ ( n ) + n S_\gamma (n)+n , which is consistent with the validity of the Riemann hypothesis.

The aim of this paper is to study generalized Hardy sums s5(h, k). By using mediants and the adjacent difference of Farey fractions, we establish a relationship between s5(h, k) and Farey fractions. Using generalized Dedekind sums and a generalized periodic Bernoulli function, we define generalized Hardy sums s5,p(h,k). A relationship between s5,p(h, k) and the Hurwitz zeta function is established. By using the definitions of Lambert series and cotπz, we establish a relationship between s5(h,k) and Lambert series.

For the generalized-Euler-constant function\n\\[\na\\mapsto \\gamma(a):=\\underset{n\\rightarrow\\infty}{\\lim}\n\\left(\\sum_{i=0}^{n-1}\\frac{1}{a+i}-\\ln\\frac{a+n-1}{a}\\right)\n\\]\ndefined on $\\R^+$, the expansion $\\gamma(a)=\\sum_{j=2}^{\\infty}\\frac{(-1)^j}{j}\\,\\zeta(j,a)$,\nwhere $\\zeta(j,a)$ is the Hurwitz zeta function, is derived and a formula for its numerical\ncomputation is presented.

It is known that the Hurwitz zeta-function $$\zeta(s,\alpha)$$ with transcendental or rational parameter $$\alpha$$ has a discrete universality property; i.e., the shifts $$\zeta(s+ikh,\alpha)$$ , $$k\in\mathbb N_0$$ , $$h> 0$$ , approximate a wide class of analytic functions. The case of algebraic irrational $$\alpha$$ is a complicated open problem. In the paper, some progress in this problem is achieved. It is proved that there exists a nonempty closed set $$F_{\alpha,h}$$ of analytic functions such that the functions in $$F_{\alpha,h}$$ are approximated by the above shifts. Also, the case of certain compositions $$\Phi(\zeta(s,\alpha))$$ is discussed.