In mathematics, the lerch zeta function, sometimes called the hurwitz lerch zeta function, is a special function that generalizes the hurwitz zeta function and the polylogarithm. After giving a brief overview of the theory of multiple zeta func tions, we derive the analytic continuation of the multiple hurwitz zeta function. The function zeta 0, z, v is often called the hurwitz zeta function or the generalized zeta function. For comparison, we summarize a pair of known series identities for the polylogarithm function, lisz, and the modi. Aspects of zetafunction theory in the mathematical works of adolf. Each integral representation allows us to obtain an analytic continuation including also a full description of poles and special values at nonpositive integers of the qanalogue of the hurwitz zeta function, and to study the classical limit of this q. The master of us all, and for riemann, one should turn to edwads riemanns zeta function to read riemanns. We give a proof based on a functional equation for the jacobi theta function z p 1 m1 e. The multiple hurwitz zeta function and a generalization of lerchs formula kamano, ken, tokyo journal of mathematics, 2006. 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. Zeros of hurwitz zeta functions american mathematical society. This thesis is brought to you for free and open access by topscholar. Two integral representations of qanalogues of the hurwitz zeta function are established. For the case of the eigenvalues of a simple quantum harmonic oscillator in one dimension 10. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number. Although these series only converge for \\res 1\, the riemann and hurwitz zeta functions are defined through analytic continuation for arbitrary complex \s \ne 1\ \s 1\ is a pole the implementation uses three algorithms. As a direct consequence we obtain a short, motivated proof of riemanns functional equation for s. In mathematics, the hurwitz zeta function, named after adolf hurwitz, is one of the many zeta functions. Hurwitz zeta function and so of dirichlet lseries and character polylogarithms. Pdf we give new integral and series representations of the hurwitz zeta function. Adamchik, derivatives of the hurwitz zeta function for rational arguments, journal of comp. The hurwitz zeta function can also be written as a series involving the gamma function, as the sum of dirichlet lfunctions 8, and for positive integer inputs, in.
Multiple hurwitz zeta functions department of mathematics and. Hurwitz zeta function, analytical number theory, functional equation. A formula for the hurwitz zeta function, riemanns functional equation and certain. A study of a certain subclass of hurwitz lerch zeta function related to a linear operator ghanim, f. Pdf a note on the hurwitz zeta function researchgate. Crandalls computation of the incomplete gamma function. It essentially generalizes an algorithm given by peter borwein for computing the riemann zeta function. An introduction to the theory of the riemann zetafunction. 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. The hurwitz zeta function is related to other special functions. Primary definition 8 formulas 19982020 wolfram research, inc. Generalized zeta function will be 0 log n a s n n s. It is possible to obtain similar formulas for r i rational numbers 27. What are zeta functions of graphs and what are they good for.
It is named after the czech mathematician mathias lerch 1. The hurwitz zeta function also has a wellstudied analytic continuation apostol 1976, and we conclude that the quantum zeta function of the perturbed oscillator has been determined, for arbitrary complex s, in terms of standard. Zeta functions, topology and quantum physics takashi. Basic zeta functions and some applications in physics. We also provide a closedform expression of the coefficients of the laurent expansion of the hurwitzzeta function about any point in the complex plane. On some integrals involving the hurwitz zeta function. Input the arguments in the fields below and hit the calculate.
The third parameter, v, can be any complex number which is not a nonpositive integer. If youre interested in eulers role in the development, i would look to dunhams euler. Riemann zeta unctionf and hurwitz zeta unctions,f which generalize the riemann zeta unction. Integrals of products of hurwitz zeta functions via feynman. We subsequently apply these to the corresponding computation of the.
Riemann zeta function hurwitz zeta function, euler summation, harmonic numbers, generalized harmonic numbers, catalan constant. Chapter 9 the functional equation for the riemann zeta. It is formally defined for complex arguments s with res 1 and q with req 0 by. Maybe can the symmetric zeta functions give new insights about the riemann hypothesis.
Such zeta functions are considered, for example, by clair and mokhtarisharghi 5aswellas grigorchuk and zuk 9. Euler sum generating functions of the forms studied in 4,2,12. Hurwitz zeta function calculator hurwitz zeta function calculator or the generalized zeta function calculator. Rigoroushighprecision computation of thehurwitzzeta. Numbertheoretic functions sage reference manual v9.
There are various methods to derive this functional equation, see e. The same as the latter, the riemann zeta function is an analytic function in the whole complex splane, except the pole at s 1. We give new integral and series representations of the hurwitz zeta function. In concentrating exclusively on the study of the riemann zeta function and its. F the main goal of this work is to approach this problem from a traditional and computational approach. Designed as a reference work and also as a graduatelevel textbook, this volume presents an uptodate and comprehensive account of the theories and applications of the various methods and techniques used in dealing with problems involving closedform evaluations of and representations of the riemann zeta function at positive integer arguments as numerous families of series associated with. Meanwhile, bailey and borwein 1 provide easilycomputable. Approximating the hurwitz zeta function mathematics.
Riemann zeta function we will eventually deduce a functional equation, relating s to 1 s. Zeros of the hurwitz zeta function in the interval 0,1. C, apart from s 1, where it has a simple pole with residue 1. The wellknown estimate of the order of the hurwitz zeta function. On the product of hurwitz zeta functions wang, nian liang and banerjee, soumyarup, proceedings of the japan academy, series a, mathematical sciences, 2017. The dirichlet l functions may be written as a linear combination of the hurwitz zeta function at rational values. Integralrepresentations of qanaloguesof the hurwitz zeta. The modified hurwitz zeta function is simply related to the hurwitz zeta function. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. For more information, see increase speed by reducing precision. The metho d dev elop ed in this pap er is illustrated b yn umerous examples of closedform ev. This series is absolutely convergent for the given values of s and q and can be extended to a meromorphic function defined for all s. The precursors to the zeta function and its development by euler and riemann especially eulers original proof to the basel problem are fantastic.
A generalized riemann zeta function known as the hurwitz zeta function can also be defined such that 14 the riemann zeta function may be computed analytically for even using either contour integration or parsevals theorem with the appropriate fourier series. Floatingpoint evaluation of the hurwitz zeta function can be slow for complex arguments or highprecision numbers. Zeta functions, lseries and polylogarithms mpmath 0. On the hurwitz zeta function with an application to the. Adolf hurwitz, zeta and lfunctions, riemann hypothesis, entire functions.
Zeta functions of graphs 3 we do not consider zeta functions of in. The zeros of this function will be symmetrically distributed around the critical line. On connection between values of riemann zeta function at. The prime number theorem, hardys theorem on the zeros of. Co ey 3 provides various convergent, recursive expressions for the hurwitz zeta function. The hurwitz zeta function has also the following integral representation. Keywords hurwitz zeta function riemann zeta function arbitraryprecision arithmetic rigorous numerical evaluation fast polynomial arithmetic power series mathematics subject classi. Series associated with the zeta and related functions. The point z1 is a simple pole for the function zeta 0, z, v. Refinement of an estimate for the hurwitz zeta function in a. The key idea in our analysis is that the hurwitz zeta function. The zeta function is an important function in mathematics.
Results about fractional derivatives of zeta functions. Rane, a new approximate functional equation for the hurwitz zeta function for rational parameter, proc. The aims of this symposium were to establish mutual understanding and to exchange ideas among researchers. This volume contains papers by invited speakers of the symposium zeta functions, topology and quantum physics held at kinki u versity in osaka, japan, during the period of march 36, 2003. We aim to investigate derivatives of zeta functions by exploring. The k ey ingredien ts in our approac h include certain kno wn in tegral represen tations for s and s. Accurately approximating the hurwitz zeta function is thus signi cant due to its important applications in quantum mechanics and in other areas of mathematics. To increase the computational speed, you can reduce the floatingpoint precision by using the vpa and digits functions. Relations among the riemann zeta and hurwitz zeta functions, as. On some families of integrals connected to the hurwitz zeta function patkowski, alexander e. 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. Higher derivatives of the hurwitz zeta function topscholar. Up to the possible existence of a siegel zero, zero free regions including and beyond the line res.
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