Riemann zeta function minus one for large positive argument, the riemann zeta function approaches one. Functional equation of the hurwitz zeta function and a. The hurwitz zeta function has a meromorphic continuation in the splane with a simple pole of residue 1 at s1. Zeros of the hurwitz zeta function in the interval 0,1. Some additions to a family of integrals related to hurwitz. Hurwitz zetafunction has no real zeroes for large negative our results are illustrated in fig. This connection is used to rephrase a modulartype transformation involving infinite series of hurwitz zeta function in terms of those involving lommel functions. 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. Error bounds for the asymptotic expansion of the hurwitz. Riemann zeta function an overview sciencedirect topics. 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. We recall here a general hurwitz lerch zeta function, which is defined in 3, 4 by the following series. We provide a probabilistic interpretation of our result by exploiting a connection between hurwitz zeta function and the cumulants of the betaexponential distribution. New result of analytic functions related to hurwitz zeta function.
The level of precision is controlled bynthe number of decimal places. On the hurwitz zeta function with an application to the. In this region the fractional part is interesting, and therefore we need a function to evaluate it explicitly. Rane, a new approximate functional equation for the hurwitz zeta function for rational parameter, proc. The integral representation is derivable from the difference of hurwitz zeta functions by substituting 25. The hurwitz zeta function coincides with riemanns zeta function. Same man who added entire gamma formula to bottom of this page 188.
We establish a series of integral formulae involving the hurwitz zeta function. 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. We obtain a new proof of hurwitzs formula for the hurwitz zeta function \ zetas, a beginning with hermites formula. Basic zeta functions and some applications in physics. Many useful and interesting properties, identities, and relations for the riemann zeta function. Pdf a note on the hurwitz zeta function researchgate. Remark on the hurwitzlerch zeta function fixed point. Furthermore, we will introduce some other zeta function which are closly related to the riemann zeta function. Keywords hurwitz zeta function riemann zeta function arbitraryprecision arithmetic rigorous numerical evaluation fast polynomial arithmetic power series mathematics subject classi. Various generalizations of the hurwitz lerch zeta function have been actively investigated by many authors. It is possible to obtain similar formulas for r i rational numbers 27.
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. The metho d dev elop ed in this pap er is illustrated b yn umerous examples of closedform ev. Certain relationships among polygamma functions, riemann zeta. 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. Important special cases of the function include, for example, the riemann zeta function, the hurwitz zeta function, the lerch zeta function, and the polylogarithm. Chapter 9 the functional equation for the riemann zeta function.
Co ey 3 provides various convergent, recursive expressions for the hurwitz zeta 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. A new class of distributions based on hurwitz zeta function. 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. We establish a series of indefinite integral formulae involving the hurwitz zeta function and other elementary and.
As a direct consequence we obtain a short, motivated proof of riemanns functional equation for s. The hurwitz zeta function has also the following integral representation. Before defining the new function, it is worth putting the family of zeta functions in perspective for our purpose. For some properties of the barnes zeta function the use of complex contour. The k ey ingredien ts in our approac h include certain kno wn in tegral represen tations for s and s. It essentially generalizes an algorithm given by peter borwein for computing the riemann zeta function. Primary definition 8 formulas 19982020 wolfram research, inc. We also show that the well known generalized gamma family of distributions is a special case of the proposed family. 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. Pdf we give new integral and series representations of the hurwitz zeta function. Symmetrization of the hurwitz zeta function and dirichlet l. This analytic continuation to c has a simple pole of residue one. Rigoroushighprecision computation of thehurwitzzeta function. Meanwhile, bailey and borwein 1 provide easilycomputable.
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. The functional equation for the hurwitz zeta function. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number. Approximating the hurwitz zeta function mathematics.
Rigorous highprecision computation of the hurwitz zeta function. We give a proof based on a functional equation for the jacobi theta function z p 1 m1 e. On some integrals involving the hurwitz zeta function. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. 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. Titchmarsh, the theory of the riemann zeta function. At rational arguments the hurwitz zeta function may be expressed as a linear combination of dirichlet lfunctions and vice versa. 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. Zeta functions and polylogarithms zeta s,a primary definition 8 formulas primary definition 8 formulas zeta. A new generalization of the riemann zeta function and its. The multiple hurwitz zeta function and the multiple hurwitz euler eta function choi, junesang and srivastava, h. On the hurwitz zeta function with an application to the beta. The hurwitz zeta function zetas,a is a generalization of the riemann zeta function zetas.
On the real zeroes of the hurwitz zetafunction and bernoulli. The zeta function is an important function in mathematics. Thanks for contributing an answer to mathematics stack exchange. On some series representations of the hurwitz zeta function. On some families of integrals connected to the hurwitz zeta function patkowski, alexander e. There are various methods to derive this functional equation, see e. Derivatives of the hurwitz zeta function for rational arguments. Applications are given to integrals of bernoulli polynomials, ln. In this paper, we present two extensions of the riemann zeta function which are closely related to. A special case of this function is the hurwitz or generalized zeta function. We study its properties and relations with other special functions. 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. Pdf the hurwitz zeta function as a convergent series. Adamchik, derivatives of the hurwitz zeta function for rational arguments, journal of comp.
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