Integral representations for certain series

On some series representations of the Hurwitz zeta function

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).

The Gamma Function and 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 Sect. 1.9 we considered summation of finite series. In practice it is often necessary to deal with infinite series which contain an infinite number of terms a n . Here a n is a general (we say “the n-th”) term of the series which is constructed via a some kind of formula depending on an integer n = 1, 2, 3, …. For instance, the rule \(a_{n} = a_{0}q^{n}\) with n = 0, 1, 2, 3, … corresponds to an infinite geometric progression $$\displaystyle{a_{0} + a_{0}q + a_{0}q^{2} + \cdots = a_{ 0}\left (1 + q + q^{2} + \cdots \,\right ).}$$

is well known that the Hurwitz zeta function ζ(s, α) with rational or transcendental parameter α is universal in the sense of Voronin, i.e., a wide class of analytic functions can be approximated by the shifts ζ(s + iτ, α), τ ∈ ℝ. The case of algebraic irrational α is still an open problem. It is proved that there exists a nonempty closed set of analytic functions that can be approximated by shifts ζ(s + iτ, α) with algebraic irrational α.

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.

A different and technical proof for a series representation of log s is provided.The mentioned series representation was used to reproduce some Bailey-Borwein-Plouffe (BBP) type formulas for a variety of numbers, including some classical BBP type formulas for π.A series representation, similar to that of log s, for log 2 (s) is also derived.

We first consider a summation procedure for some trigonometric series in terms of the Riemann zeta and related functions. In some cases these series can be brought in closed form, which means that the infinite series are represented by finite sums. Afterwards, we show some applications of our results to the summation of series involving Bessel or Struve functions. Further, relying on results from the previous sections, we obtain sums of series involving a Bessel or Struve integral. These series are also represented as series in terms of the Riemann zeta and related functions of reciprocal powers and can be brought in closed form in certain cases as well. By replacing the function appearing in a Bessel and Struve integral with particular functions, we find sums of new series.

The circular cross-section problem is the problem that is normally solved in the literature, both because of its inherent importance and because it is an excellent example of the use of Bessel functions. Most of these solutions are aimed at the circuit viewpoint, i.e. resistance, reactance and impedance, rather than power loss, which is the purpose of direct resistance heating (DRH). A.G. Warren ("Mathematics Applied to Electrical Engineering", Chapman and Hall, London, 1949) has an excellent section (pp. 243ff.) on current in a circular conductor but the book has long been out-of-print; also, it uses unrationalised CGS concepts, so we shall start again from first principles. For the reader not conversant with Bessel functions, we do not make a lot of fuss about them here: suffice it to say that they are the equivalent in a cylindrical world of the sinusoids and exponentials of the previous chapters and, like them, are described by infinite series. We find that there is a close similarity between the solutions for the slab and those for the cylinder. Just as, in the thin slab, we found that the two sides interacted to give hyperbolic functions, Bessel functions are needed because every point in the section of a circular conductor is affected by the currents flowing in the rest of the conductor.

This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws.

In the author's paper [7] it was proved that the fractional parts of the partial sums of an infinite series (of real terms) diverging to +∞ or −∞, in which the general term tends to zero, are everywhere dense in the closed unit interval. This result was extended to series of infinite oscillation (see Remark 4.1 of the said paper) on the argument that a sequence of partial sums having infinite oscillation has a subsequence that diverges to +∞ or −∞.

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.