Review of the Book

Series Associated with the Zeta and Related Functions

By

H.M. Srivastava and Junesang Choi

 

Kluwer Academic Publishers, Dordrecht/Boston/London, 2001, ix,

pp. 388, ISBN 0-7923-7054-6

The Riemann Hypothesis (RH) concerning the zeros of the Riemann Zeta function has had, and is continuing to have, a profound and far-reaching effect on the activity of many mathematicians since its enunciation by Riemann in 1859. At the 1900 address to the Paris International Congress of Mathematicians, Hilbert listed the RH as one of his famous list of 23 mathematical problems. It still remains unsolved to this day despite the effort expounded to date. In the quest for its solution it has developed many new areas of mathematics because of its implication in the solution of many problems (see for example, Conrey (2003) and Borwein et al. (2000)).

The current book investigates an even more fundamental problem, that relating to the summation of reciprocal powers of series which are broadly termed as Zeta and related functions. The book aims, with considerable success, in giving a unified and systematic treatment of identities involving the Riemann zeta function and related functions including; the Hurwitz zeta and Lerch's transcendent .

Some of the results stem from a 1729 theorem of Christian Goldbach as reported in some correspondence with Daniel Bernoulli. The theorem involves the closed form solution of

(1)

where the sum is over omega belonging to the set of all non-trivial integer kth powers. In terms of the Riemann zeta function , Goldbach's theorem may be restated as

(2)

where indicates the fractional part of

The book is aimed at providing a source of reference material for researchers requiring results associated with the Zeta function and related series. Detailed and methodical presentation of techniques for handling zeta type series is given and placed in a historical context giving the reader an appreciation of its origins. An extensive list of references is provided together with tables of identities demonstrating the wealth and depth of many results of series and integrals satisfied by Zeta related functions.

The book consists of six chapters each of which culminates with generous sets of problems relating to the matter covered in the chapters and reference is made to the original source of the problems from which the interested reader may seek further elaboration on the solution methodology.

The first chapter provides a very nice overview of some classical functions and their properties which gives good background material required for the subsequent chapters. It covers; Gamma, Beta, Polygamma and Hypergeometric functions together with Bernoulli, Euler numbers and polynomials and, Stirling numbers of the first and second kind. Chapter 2 defines the zeta and related functions and develops results related to those in Chapter 1 while in Chapter 3 series involving the zeta function are evaluated giving a very extensive list of result while in Chapter 4 the emphasis is in providing representations in terms of fast convergent series and investigating convergence properties of Zeta functions in particular for the odd integers. In Chapter 5, the computation of the determinants of the Laplacians on manifolds of constant curvature is investigated which involves the closed form evaluation of Zeta functions. The final Chapter 6 involves the investigation of closed form evaluation of some of the special functions defined in Chapter 1 in terms of Zeta related functions.

I have a concern with the book and that is with the method of referencing of equations in sections such as a.b(c) referring to equation c in section a.b of chapter a. The equation numbering is started afresh in each section although the top of the pages only indicate the chapter so that one needs to find the start of the section by going to the table of contents.

Overall, the book will serve as a very valuable reference presenting and documenting an extensive array of both methodologies and results involving the Zeta and related functions. It is a worthwhile addition to the arsenal for researchers interested in this area of analysis which has a very long historical interest both in itself and in the extensive areas of application.

 

P. Cerone

 

[1] J.M. BORWEIN, D.M. BRADLEY and R.E. CRANDALL, Computational strategies for the Riemann zeta function, Numerical analysis in the 20th Century, Vol. I, Approximation theory. J. Comput. Appl. Math., 121(1-2) (2000), 247--296.

[2] J. BRIAN CONREY, The Riemann hypothesis, Notices Amer. Math. Soc., 50(3) (2003), 341--353.