Uni ed algorithms for polylogarithm, lseries, and zeta variants richard e. It approximates the trivial and nontrivial zeros of the riemann zeta function. We have introduced a new generalization of the riemann zeta function. Some generalizations of the riemann zeta function are outlined, as well as the calculation of zeta constants and. The polylogarithm is a special case of the incomplete polylogarithm function.
Notes on the riemann zeta function january 25, 2007 1 the zeta function 1. Some properties of the generalized riemann zeta function are investigated. Information about the ramified primes and about the behavior at infinity is contained in three integers. Polylogarithms and physical applications cristian vergu july 10, 20 contents.
Riemann zeta function an overview sciencedirect topics. If the second argument is 0, then the polylogarithm is equal to 0 for any integer value of the first argument. In this case one expects to have a product formula in terms of local factors. Polylogarithms, dedekind zeta functions, and the algebraic. Furthermore, we will introduce some other zeta function which are closly related to the riemann zeta function. Moreover, in 1859 riemann gave a formula for a unique the socalled holomorphic extension of the function onto the entire complex plane c except s 1. A new generalization of the riemann zeta function and its.
New integral representations of the polylogarithm function. In section 7 we turn to another theme, the relation of special zeta values to invariants of moduli spaces of various kinds. Crandall1 march 26, 2012 in memory of gentle colleague jerry keiper 19531995 abstract. Uni ed algorithms for polylogarithm, lseries, and zeta. Many of the chapters and sections of the book have been significantly modified or rewritten, and a new chapter on the theory and applications of the basic or q. Its euler product tells how the unramified primes of q split in f. Zeta functions, lseries and polylogarithms mpmath 0. Matlab live scripts support most mupad functionality, although there are some differences. Riemanns zeta function if s 1 is a real number, then the series s x n 1 1 ns converges. The polylog function has special values for some parameters. Although these series only converge for, the riemann and hurwitz zeta functions are defined through analytic continuation for arbitrary complex is a pole the implementation uses three algorithms. Cameron franc special values of riemanns zeta function.
Multiple zeta functions, multiple polylogarithms and their. The riemann zeta function is defined as the analytic continuation of the function defined for. Zeta and qzeta functions and associated series and integrals. Realizing that their plagiaristic actions risked running afoul of the mathematical community, the clay math institute felt compelled to make a preemptive peace o ering or, as it is known in.
Riemann zeta function, analytic properties, definitions, polylogarithms, relations to other functions, series expansions notes. The exposition is selfcontained, and required a preliminary knowledge of only the ele. The polylogarithm function is equivalent to the hurwitz zeta function either function can be expressed in terms of the other and both functions are special cases of the lerch transcendent. Here we give a rapidly converging continuedfraction expansion of nforany integer n 2. A special case of this function is the hurwitz or generalized zeta function. Zeta 198 formulas zeta s,a 321 formulas riemannsiegelz 47 formulas lerchphiz,s,a 273 formulas ramanujantaul 16 formulas ramanujantauz formulas zeta related functions. The fact that this function is analytic in this region of the complex. Zeta functions in algebraic geometry mircea mustat.
We avoid asymptotic expansions in favor of manifestly. Both dirichlet eta function and riemann zeta function are special cases of polylogarithm. This thesis is an exposition of the riemann zeta function. What are zeta functions of graphs and what are they good for. Dedekind zeta functions of arbitrary number fields cf. The algorithm provides a rapid means of evaluating liszfor general values of complex s and the region of complex z. Kudryavtseva 1 filip saidak peter zvengrowski abstract an exposition is given, partly historical and partly mathematical, of the riemann zeta function s and the associated riemann hypothesis. By a global zeta function we mean an object that takes into account all primes. Included are techniques of analytic continuation and relationships to special functions.
An efficient algorithm for accelerating the convergence of. A special case of our generalization converges locally uniformly to the riemann zeta function in the critical strip. Dynamical zeta functions mark pollicott contents 1 introduction 2. As such, it may be taken as an extension of the techniques given by borweins an ef. 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. Buy multiple zeta functions, multiple polylogarithms and their special values number theory and its applications on free shipping on qualified orders. This function is going to be used in the last section where we are going to talk about the symmetric zeta function, which is based on a linear combination of lfunctions and zeta functions. If the second argument is 1, then the polylogarithm is the riemann zeta function of the first argument. Using techniques similar to those of riemann, it is shown how to locate and count nontrivial zeros of s. Zeta functions and polylogarithms 1,571 formulas zeta functions. In computer science, polylogarithmic functions occur as the order of time or memory used by some algorithms e. Zeta functions and polylogarithms polylognu,z series representations. The chapter also provides an overview of the polylogarithm functions, which include the legendres chi function, clausens integral or clausens function, and the hurwitzlerch zeta function.
The zeta function is an important function in mathematics. While the dirichlet series expansion for the eta function is convergent only for any complex number s with real part 0, it is abel summable for any complex number. This zeta function is an in nite product over prime ideals pin o k, the ring of algebraic integers of k. Zeta function, dilogarithm and polylogarithm functions. This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to. Fs of an algebraic number field f is the most important invariant of f. We describe a general computational scheme for evaluation of a wide class of numbertheoretical functions. It is therefore natural to ask for similar realanalytic and singlevalued modification of the higher polylogarithm functions li. The polylogarithm and the hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other. Alternatively, the hurwitz zeta can be very rapidly evaluated by means of an eulermaclaurin series. This is a special case of a more general expansion which we.
The eulermaclaurin series isa clear performance winner for the hurwitz zeta, while the borwein algorithm is superior for evaluating the polylogarithm. A polylogarithmic function in n is a polynomial in the logarithm of n. It essentially generalizes an algorithm given by peter borwein for computing the riemann zeta function. Polylogarithmic zeta functions and their padic analogues. The basic example is the wellknown factorization of the riemann zeta function. In section 6 we return to critical values to give one or two examples of special values of zeta functions at the symmetry point of their functional equation. The prime number theorem, hardys theorem on the zeros of. However, the formula 2 cannot be applied anymore if the real part. Riemannsiegeltheta 103 formulas ramanujantautheta 156 formulas polylogarithms. Zeta 198 formulas zetas,a 321 formulas riemannsiegelz 47 formulas lerchphiz,s,a 273 formulas ramanujantaul 16 formulas ramanujantauz formulas zetarelated functions. Zeta and qzeta functions and associated series and integrals is a thoroughly revised, enlarged and updated version of series associated with the zeta and related functions. The polylogarithm is related to the complete fermidirac integral as.
1384 1385 1171 1478 1464 991 336 646 824 633 1006 1286 537 276 1323 1556 1263 1178 187 727 1042 1439 545 633 135 865 1566 614 1297 436 147 1283 162 317 1473 1088 645 1017 935 1474 63 494 1036 1241 752 1038 427