On some series representations of the Hurwitz zeta function

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.

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 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.

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.

The objective of present research is to introduce and analyze a new special function that occurs as a part of the kernel of the integral representation of the λ-generalized Hurwitz-Lerch zeta functions. One important aspect of the analysis of special functions is to study their properties. It is found that λ-generalized gamma function satisfies the log convex and derivative properties. Recurrence relation and reflection formula are achieved that are always important to study the behavior of new functions. As an application, a new series representation of the λ-generalized Hurwitz-Lerch zeta functions is established. These new results are validated by analyzing their important cases that agree with the known results. It is interesting to note that the coefficients in the series representation of the family of zeta functions are generalized from “1” to “gamma function” and then to “generalized gamma” and “λ-generalized gamma functions” in a simple and natural way.

This article explores the connection between Fourier series and various zeta functions, including the Riemann zeta function and its generalizations. Specifically, we derive recurrence formulas for even and odd values of zeta functions using Fourier expansions, extending these results to the Hurwitz zeta function.

Five basic mistakes are found in the Riemann’s original paper proposed in 1859. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane and in the domain of the function, when the left-hand side of equation is finite, the right-hand side may be infinite, and vice versa. The Riemann Zeta function equation holds only at the point Re(s) = 1/2(s= a+ \(i b\)) . However, at this point, the Zeta function is infinite, rather than zero, to make Riemann hypothesis untenable. 2. An integral item around the original point of coordinate system was ignored when Riemann deduced the integral form of Zeta function. The item was convergent when Re(s) > 1but divergent when Re(s) < 1 .The integral form of Zeta function does not change the divergence of its series form. 3. A summation formula was used in the deduction of the integral form of Zeta function. The applicative condition of this formula is x > 0 . At point x = 0 , the formula is meaningless. However, the lower limit of Zeta function’s integral is x = 0 , so the formula can not be used. 4. Because the integral lower limit of Zeta function is zero, the integrand function is not uniformly convergent, so integral sign and sum sign can not be exchanged. But Riemann made them interchangeable, resulting in that the integral form of Zeta function is untenable. 5.The formula \(\theta(x)=\sqrt{x} \theta(1 / x)\) of Jacobi function was used to prove the symmetry of Zeta function equation. The applicable condition of this formula is also x > 0 . Because the lower limit of integral in the deduction was x = 0 , this formula can not be used too. The zero calculation of Riemann Zeta function is discussed at last. It is pointed out that because approximate methods are used, the analytic property of the original function is destroyed and the Cauchy-Riemann equation can not be satisfied. So they are not the real zeros of strict Riemann Zeta function.

This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators $\{ T_m: \, m \ge 1\}$ given by $T_m(f)(a, c) = \frac{1}{m} \sum_{k=0}^{m-1} f(\frac{a+k}{m}, mc)$ acting on certain spaces of real-analytic functions, including Lerch zeta functions for various parameter values. It determines the action of various related operators on these function spaces. It characterizes Lerch zeta functions (for fixed $s$ in the following way. It shows that there is for each $s \in {\bf C}$ a two-dimensional vector space spanned by linear combinations of Lerch zeta functions is characterized as a maximal space of simultaneous eigenfunctions for this family of Hecke operators. This result is an analogue of a result of Milnor for the Hurwitz zeta function. We also relate these functions to a linear partial differential operator in the $(a, c)$-variables having the Lerch zeta function as an eigenfunction.

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.

This thesis deals with two properties of arithmetic functions, namely, factorization and independence. In 1984, Rearick pointed out that the ring of arithmetic functions is a unique factorization domain but is not a principal ideal domain and so is not a Euclidean domain. Without the Euclidean algorithm, the problem of factorizing arithmetic functions becomes quite difficult. In the first part of this thesis, we propose a technique of factorizing certain classes of arithmetic functions and prove some results about factorization which are based mainly on the norms of such functions. The technique is a generalization of the original works of Rearick which consists of solving a special system of linear differential equations whose coefficients are polynomials in the function to be factorized. The solutions of this system are proved to be the sought after factors with increasing norms. Examples illustrating the technique are also given. In 1986, Shapiro and Sparer made an extensive study of algebraic independence of Dirichlet series. Since the ring of Dirichlet series is isomorphic to that of arithmetic functions, the study in one setting is then equivalent to the other. Shapiro and Sparer's investigation began with a theorem asserting that Dirichlet series are algebraically independent if their Jacobian does not vanish, which is classical in the case of real-valued functions. Taking the Riemann zeta function as a building block, they discovered that Dirichlet series algebraically dependent on the zeta function can uniquely be represented as power series in the logarithms of zeta function. A number of algebraic dependence results of Dirichlet series were derived as consequences. Shapiro and Sparer then went on to investigate analogous results for formal generalized Dirichlet series. Results in the second part of this thesis either extend or simplify some of Shapiro and Sperer's results. These include, for example, replacing the zeta function by Dirichlet series with completely multiplicative coefficients to obtain similar log-series expansions, dependence of series with infinite support, and dependence of non-units whose norms are relatively prime.

In this paper we generalize, to an arbitrary number field, the theorem which gives the probability that two integers are relatively prime. The probability that two integers are relatively prime is 1/ζ(2), where ζ is the Riemann zeta function and 1/ζ(2)=6/π 2. The theorem for an arbitrary number field states that the probability that two ideals are relatively prime is the reciprocal of the zeta function of the number field evaluated at two. In particular, since the Gaussian integers are an unique factorization domain, we get the probability that two Gaussian integers are relatively prime is 1/ζ G (2) where ζ G is the zeta function associated with the Gaussian integers. In order to calculate the Gaussian probability, we use a theorem that enables us to factor the zeta function into a product of the Riemann zeta function and a Dirichlet series called an L-series. For the Gaussian integers we get: ζ G (2)=ζ(2)L(2,χ), where $$L\left( {2,\chi } \right) = 1 - \frac{1}{{3^2 }} + \frac{1}{{5^2 }} - \frac{1}{{7^2 }} + \cdots .$$ We use this factorization to approximate the Gaussian probability to 17 decimal places.

Bernoulli numbers appear as special values of zeta functions at integers and identities relating the Bernoulli numbers follow as a consequence of properties of the corresponding zeta functions. The most famous example is that of the special values of the Riemann zeta function and the Bernoulli identities due to Euler. In this paper we introduce a general principle for producing Bernoulli identities and apply it to zeta functions considered by Shintani, Zagier and Eie. Our results include some of the classical results of Euler and Ramanujan. Kummers congruences play important roles in the investigation of p -adic interpolation of the classical Riemann zeta function.

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.

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).