The hurwitz zeta function has a meromorphic continuation in the splane with a simple pole of residue 1 at s1. On the hurwitz zeta function with an application to the beta. The hurwitz zeta function coincides with riemanns zeta function. Hurwitz zetafunction has no real zeroes for large negative our results are illustrated in fig. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. We prove a monotonicity property of the hurwitz zeta function which, in turn, translates into a chain of inequalities for polygamma functions of different orders. A special case of this function is the hurwitz or generalized zeta function. The hurwitz zeta function has also the following integral representation. Symmetrization of the hurwitz zeta function and dirichlet l. Some additions to a family of integrals related to hurwitz. Remark on the hurwitzlerch zeta function fixed point. The hurwitz zeta function zetas,a is a generalization of the riemann zeta function zetas. Riemann zeta function we will eventually deduce a functional equation, relating s to 1 s.
It essentially generalizes an algorithm given by peter borwein for computing the riemann zeta function. The k ey ingredien ts in our approac h include certain kno wn in tegral represen tations for s and s. On the hurwitz zeta function with an application to the. Functional equation of the hurwitz zeta function and a. Rigoroushighprecision computation of thehurwitzzeta function. This analytic continuation to c has a simple pole of residue one. New result of analytic functions related to hurwitz zeta function. Zeta functions and polylogarithms zeta s,a primary definition 8 formulas primary definition 8 formulas zeta. We give a proof based on a functional equation for the jacobi theta function z p 1 m1 e. The integral representation is derivable from the difference of hurwitz zeta functions by substituting 25. This connection is used to rephrase a modulartype transformation involving infinite series of hurwitz zeta function in terms of those involving lommel functions. In this paper, i will demonstrate an important fact about the zeros of the zeta function, and how it relates to the prime number theorem.
In this region the fractional part is interesting, and therefore we need a function to evaluate it explicitly. Applications are given to integrals of bernoulli polynomials, ln. Jun 30, 2011 here we provide a new generalization of the riemann zeta function that is also related to the efd and ebe functions and to the hurwitz lerch zeta function. Error bounds for the asymptotic expansion of the hurwitz. A new generalization of the riemann zeta function and its. Accurately approximating the hurwitz zeta function is thus signi cant due to its important applications in quantum mechanics and in other areas of mathematics.
We obtain a new proof of hurwitzs formula for the hurwitz zeta function \ zetas, a beginning with hermites formula. On some series representations of the hurwitz zeta function. Pdf a note on the hurwitz zeta function researchgate. Riemann zeta function an overview sciencedirect topics.
In this paper, we present two extensions of the riemann zeta function which are closely related to. Very recently, srivastava presented a systematic investigation of numerous interesting properties of some families of generating functions and their partial sums which are associated with various classes of the extended hurwitz lerch zeta functions. Keywords hurwitz zeta function riemann zeta function arbitraryprecision arithmetic rigorous numerical evaluation fast polynomial arithmetic power series mathematics subject classi. The metho d dev elop ed in this pap er is illustrated b yn umerous examples of closedform ev. Theorem1and the derived inequalities in 3 add to the current body of literature about inequalities and monotonicity properties of the hurwitz zeta function 1,36 and polygamma functions 711. We study its properties and relations with other special functions. Meanwhile, bailey and borwein 1 provide easilycomputable. Pdf we give new integral and series representations of the hurwitz zeta function. Approximating the hurwitz zeta function mathematics. Many useful and interesting properties, identities, and relations for the riemann zeta function. Primary definition 8 formulas 19982020 wolfram research, inc. Co ey 3 provides various convergent, recursive expressions for the hurwitz zeta function. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number.
A new class of distributions based on hurwitz zeta function. It is possible to obtain similar formulas for r i rational numbers 27. Various generalizations of the hurwitz lerch zeta function have been actively investigated by many authors. Important special cases of the function include, for example, the riemann zeta function, the hurwitz zeta function, the lerch zeta function, and the polylogarithm. We provide a probabilistic interpretation of our result by exploiting a connection between hurwitz zeta function and the cumulants of the betaexponential distribution. Zeros of the hurwitz zeta function in the interval 0,1.
Before defining the new function, it is worth putting the family of zeta functions in perspective for our purpose. The functional equation for the hurwitz zeta function. In the next section we provide a new family of distributions, the zeta family, which is based on the hurwitz zeta function, and provide its characteristic function. We also provide a closedform expression of the coefficients of the laurent expansion of the hurwitz zeta function about any point in the complex plane. We also show that the well known generalized gamma family of distributions is a special case of the proposed family. Basic zeta functions and some applications in physics.
Derivatives of the hurwitz zeta function for rational arguments. There are various methods to derive this functional equation, see e. We recall here a general hurwitz lerch zeta function, which is defined in 3, 4 by the following series. Certain relationships among polygamma functions, riemann zeta. Furthermore, we will introduce some other zeta function which are closly related to the riemann zeta function.
Thanks for contributing an answer to mathematics stack exchange. Here, we aim at giving certain presumably new and potentially useful relationships among polygamma functions, riemann zeta function, and generalized zeta function by modifying chens method. At rational arguments the hurwitz zeta function may be expressed as a linear combination of dirichlet lfunctions and vice versa. Rigorous highprecision computation of the hurwitz zeta function. Using common mathematical packages such as mathematica, we can calculate the moments of the zeta family immediately using the hurwitz zeta function nzetas,r,n and the gamma functionna,n. The purpose here is to motivate the reader to study mellin transforms and their applications in the evaluation of harmonic sums as well as verify and perhaps improve a. On some families of integrals connected to the hurwitz zeta function patkowski, alexander e. Rane, a new approximate functional equation for the hurwitz zeta function for rational parameter, proc. We establish a series of integral formulae involving the hurwitz zeta function. We establish a series of indefinite integral formulae involving the hurwitz zeta function and other elementary and. Titchmarsh, the theory of the riemann zeta function.
On some integrals involving the hurwitz zeta function. As a direct consequence we obtain a short, motivated proof of riemanns functional equation for s. On the real zeroes of the hurwitz zetafunction and bernoulli. Adamchik, derivatives of the hurwitz zeta function for rational arguments, journal of comp. Chapter 9 the functional equation for the riemann zeta function.
The characteristic function of the pdf 3 takes the following form. In this paper, several relations are obtained among the riemann zeta and hurwitz zeta functions, as well as their products. For some properties of the barnes zeta function the use of complex contour. Yeah, im jealous the riemann hypothesis is named after the fact that it is a hypothesis, which, as we all know, is the largest of the three sides of a right triangle. Same man who added entire gamma formula to bottom of this page 188. Pdf the hurwitz zeta function as a convergent series. The level of precision is controlled bynthe number of decimal places.