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Zentralblatt MATH Database 1931 – 2007

c 2007 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag

1076.11031

Jorgenson, Jay; Lang, Serge

Pos

n

(R) and Eisenstein series. (English)

Lecture Notes in Mathematics 1868. Berlin: Springer. viii, 168 p. EUR 29.95/net; sFr.

54.50; £23.00; $ 44.95 (2005). [ISBN 3-540-25787-X/pbk]

http://dx.doi.org/10.1007/b136063

This book treats a family of Dirichlet series in several variables associated with positive

definite quadratic forms; they were introduced by A. Selberg in 1959 [see Number the-

ory, trace formulas and discrete groups, Symp. in Honor of Atle Selberg, Oslo/Norway

1987, 467-484 (1989; Zbl 0675.10030), pp. 473-474]. Although they are regarded to-

day primarily as Eisenstein series associated with general linear groups Selberg em-

phasized that they were generalizations of Epstein’s zeta function and appears to have

expected that they would also have direct arithmetical applications in the study of

quadratic forms. Selberg only sketched the main results and has not since returned to

this topic. The subject was taken up by R. P. Langlands [On the functional equations

satisfied by Eisenstein series. Lecture Notes in Mathematics. 544. (Berlin: Springer)

(1976; Zbl 0332.10018), Appendix 1 which apparently dates from 1962], by H. Maaß

[Sem. Delange-Pisot-Poitou 12 (1970/71), Theorie Nombres, No. 22, 18 p. (1972; Zbl

0244.10028), published in 1971] and by A. Terras [Nagoya Math. J. 42, 173-188 (1971;

Zbl 0212.07702), and Nagoya Math. J. 44,89-95 (1971; Zbl 0212.07801), also published

in 1971]. This method consists of several steps. To define the functions in question of

all one needs the basic reduction theory of quadratic forms and the theory of eigenfunc-

tions of the algebra of invariant operators, a theory mainly due to S. Bochner. These

are applied to certain theta series which are sums over a lattice. Selberg gave a novel

interpretation of a transformation used by Riemann in his paper on prime numbers to

show that one could remove terms where the determinant, a polynomial function on

this lattice, is zero by the application of a differential operator. This means that one

could apply a Mellin transform to represent the Dirichlet series in question. This repre-

sentation does not quite suffice to give the full analytic continuation with the functional

equations. Selberg noted that it could be supplemented by an application of Bochner’s

Tube Theorem to achieve these ends. (Langlands, who carries out this argument from

first principles, refers to this as “ involving a form of Hartog’s [sic] lemma”. He also in-

dicates (loc.cit., p. iv) that Godement also found the same arguments independently of

Selberg, but his work does not seem to have been published.) The purpose of the book

under review is to give a crisp version of this theory. It is developed ad ovo – in other

words all of the necessary reduction theory and harmonic analysis is developed simply

and efficiently. Although some of the details are, by their nature, intricate, this book

reduces these to a minimum and provides an accessible account to this important and

beautiful area of mathematics. Although the standard methods for treating Eisenstein

series today replace theta series by the functional-analytic methods (such as that due

to Roelcke and Selberg, as in Langlands’ book quoted above), Langlands’ admonition,

that this method “should not be forgotten” (loc.cit., p. iv) remains just as valid now

as it was thirty years ago. Indeed it has been experiencing a renaissance recently and

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promises to be fruitful in related areas. One final word - the reviewer received this

book for review almost simultaneously with the sad news of the death of Serge Lang

(12th September, 2005). The loss of this unique figure in the mathematics of the last

half-century has made the preparation of this review a poignant task.

Samuel James Patterson (Gottingen)

Keywords : theta series; Eisenstein series; general linear group; reduction theory; har-

monic analysis; Bochner tube theorem

Classification :

∗11F55 Groups and their modular and automorphic forms (several variables)

43A85 Analysis on homogeneous spaces

14K25 Theta-functions

32A50 Harmonic analysis of several complex variables

Cited in ...

0729.11001

Selberg, Atle

Collected papers. Volume II. (English)

Berlin etc.: Springer-Verlag. viii, 251 p. DM 138.00/hbk (1991). [ISBN 3-540-50626-8]

Vol. I (Springer 1989) of these Collected papers was reviewed in Zbl 0675.10001.

Vol. II under review contains several articles which over the years had remained unpub-

lished but whose content formed (at least part of) the subject of lectures which were

given by the author at various times and various places. Bearing in mind the author’s

well-known highly condensed style it would be a hopeless enterprise to try to collect the

contents of the present volume in the nutshell of a brief review. Hence we must restrict

to some rough indications.

The work under review contains four articles of quite different size; following the num-

bering of vol. I these bear the numbers 42-45.

No.42. Linear operators and automorphic forms (pp. 1-13). Let G be the group

of analytic mappings of a bounded symmetric complex domain B onto itself. The

author studies linear operators on functions defined on B which have the property of

transforming with a multiplier on each side under the maps of G. The problem is, for

which B and which multipliers do such operators exist and how to determine them

explicitly. This problem is solved by means of the known classification of irreducible

bounded domains and by means of Koecher’s theory of homogeneous positivity domains.

No.43. Remarks on the distribution of poles of Eisenstein series (pp. 15-45). Let Γ <

PSL

2

(

R

) be a cofinite discrete group and χ a one-dimensional unitary representation

on Γ with κ

1

> 0 singular cusps. As is well known the number N

χ

(T) of eigenvalues <

1/4+T

2

of the hyperbolic Laplacian has an asymptotic law which envolves the variation

of the argument of the determinant φ (s,χ) of the scattering matrix. This contribution

is notoriously difficult to handle. The problem of estimating the contribution of φ (s,χ)

is essentially equivalent to the problem of estimating the number of zeros ρ = β + iγ of

φ (s,χ) in the region β ≥ 1/2, 0 ≤ γ ≤ T, and the article under review gives a notable

contribution to this problem which witnesses to the author’s consummate mastership of

the analytic machinery in the area of zeta functions. The author states a conjecture to

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the effect that the congruence subgroups of the modular group have the largest discrete

spectrum (given µ (Γ \ H) and κ

1

).

No.44. Old and new conjectures and results about a class of Dirichlet series (pp. 47-

63). These are notes of lectures presented at the Amalfi Conference on Number Theory,

1989. The paper deals with investigating the distribution of the values of log F(s) near

the critical line for a class of Dirichlet series with functional equation and with the

asymptotic behaviour of various sums associated with the coefficients of such Dirichlet

series. There are various stimulating conjectures (which are related to several other con-

jectures like the Sato-Tate conjecture, Langlands conjectures, Riemann conjecture...).

In addition, there are results and conjectures on the zeros of F(s) − a for a = 0, and

the case of a linear combination F(s) of Dirichlet series with functional equation and

Euler product receives special attention. Concluding remark of the author: “A more

complete account with proofs is under preparation and will in time appear elsewhere.”

No.45. Lectures on sieves (pp. 65-247). This article of book-length develops the general

theory for variable than just constant sifting density. The author gives a full account of

his theory of the (Brun-) Buchstab-Rosser sieve. (The results are of course in agreement

with those independently found by H. Iwaniec.) In addition, there are more details

on the Λ

2

-sieve and the Λ

2

Λ

−

-sieve. The author gives some illuminating historical

digressions and comments on important examples. There are beautiful applications,

e.g. (p. 232)

π(x + y) − π(x) <

2y

log y + 2,8

for y > x

0

and all x > 0 (Brun-Titchmarsh), and an “early approach” to the twin

prime and Goldbach problem is described. The article under review is a comprehensive

account of work which was surveyed by the author on various earlier occasions (see e.g.

Vol. I, No.40; Zbl 0675.10030).

This volume by one of the master mathematicians of our time may well prove to be a

gold mine of inspiration for future research.

J.Elstrodt (Munster)

Keywords : variable sifting density; Brun-Buchstab-Rosser sieve; automorphic forms;

multipliers; bounded domains; homogeneous positivity domains; poles of Eisenstein

series; hyperbolic Laplacian; congruence subgroups; discrete spectrum; Dirichlet series

with functional equation; Euler product

Classification :

∗11-03 Historical (number theory)

01A75 Collected or selected works

32-03 Historical (several complex variables and analytic spaces)

11M06 Riemannian zeta-function and Dirichlet L-function

11M41 Other Dirichlet series and zeta functions

11N35 Sieves

11N36 Appl. of sieve methods

11F72 Spectral theory

32N05 General theory of automorphic functions of several complex variables

32N15 Automorphic functions in symmetric domains

Cited in ...

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Zentralblatt MATH Database 1931 – 2007

c 2007 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag

0675.10001

Selberg, Atle

Collected papers. Volume 1. (German)

Berlin etc.: Springer-Verlag. vi, 711 p. DM 228.00 (1989). [ISBN 3-540-18389-2]

The publication of the collected papers of Atle Selberg, receipient of the 1950 Fields

Medal and the 1986 Wolf Prize in Mathematics, will be highly welcomed by the math-

ematical community. This first volume contains all the mathematical papers published

by A. Selberg in mathematical journals and congress proceedings, and it contains a set

of lecture notes which was hitherto unpublished. Volume II (to appear) will contain

material on which A. Selberg has lectured at different times and places in recent years

but which has not so far appeared in print. The present collection of all the printed

articles in one handsomely produced volume is particularly valuable because many of

Selberg’s papers originally appeared in journals and congress proceedings which are not

available in many mathematical libraries.

Selberg started his publication activities with some papers on mock theta functions,

modular forms and Dirichlet series, and on the theory of functions of a complex variable.

In these works he e.g. developed the theory of Poincare series and the metrization of

the space of cusp forms, and he invented the so-called Rankin-Selberg method. Great

highlights of Selberg’s oeuvre are his deep researches on the distribution of the zeros

of the Riemann zeta-function and the Dirichlet L-functions, his elementary proofs of

the prime number theorem and the prime number theorem for arithmetic progressions,

and his version of the sieve method. The address of H. Bohr in the Proceedings of the

International Congress of Mathematicians 1950 gives a sense of the great impact which

these advances made on the contemporaries.

These researches are supplemented by the joint work with S. Chowla on Epstein’s zeta

functions culminating in the Chowla-Selberg formula. In the fifties, Selberg started

his epoch-making work on harmonic analysis and discontinuous groups culminating in

the Selberg Trace Formula and in the introduction of the Selberg zeta-function, and he

initiated his work on the rigidity of cocompact and cofinite groups.

In paper No.33 on the estimation of Fourier coefficients of modular forms we also find

the Linnik-Selberg conjecture and Selberg’s ingenious proof of the estimate λ

1

≥ 3/16

for the smallest positive eigenvalue of the Laplacian on congruence subgroups of the

modular group. (The latter bound has not been generally improved to the present day,

to the best of my knowledge.)

Several of the more recent publications of A. Selberg deal with sifting problems. Much

of his research in this area was done already in the 1950’s, but it was only in the 1970’s

that the author started to publish several papers on this subject. One of these, paper

No.40 of the present volume, is his contribution to a mathematical symposium held at

the University of Oslo, June 14-20, 1987, to celebrate the 70th birthday of Atle Selberg.

(Since full bibliographical data for this paper are missing on p. 709, we give them here:

40. Sifting problems, sifting density and sieves. Number Theory, Trace Formulas and

Discrete Groups. Symposium in Honor of Atle Selberg, Oslo, Norway, July 14-21, 1987,

ed. by K. E. Aubert, E. Bombieri, D. Goldfeld, pp. 467-484, Academic Press, New

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York- London 1989; see the review 10030 in this Zbl. volume.)

No.39 of the present volume is the set of lecture notes on Harmonic Analysis, Part

II, Gottingen, summer semester 1954. In this work, written down in 1955, the author

discusses the eigenvalue problem of automorphic forms for non-cocompact cofinite sub-

groups of SL(2,

R

). The main part of these notes is devoted to the analytic continuation

of the Eisenstein series. Moreover it is sketched how the Eisenstein series bring in new

terms into the trace formula, the analogue of Weyl’s law is developed (with a hint at

the now so-called Roelcke- Selberg conjecture), and the properties of the Selberg zeta-

function are sketched. In addition, there is an introduction to the Gottingen lecture

notes in which Selberg comments on the history of the manuscript and on subsequent

improvements of the method. Copies of the Gottingen lecture notes have been in clan-

destine circulation for 34 years and it is a great pleasure to see these notes in print.

The technical quality of the book under review is that of the well-known blue series

of collected papers published by Springer-Verlag. Some of the papers are followed by

useful comments by A. Selberg. Unfortunately, typographical errata in the original

papers have not been corrected in the present edition.

This volume should be in the library of every mathematical department, and many

mathematicians will want to add it to their personal library.

J.Elstrodt

Keywords : Collected works; discrete group; modular forms; Dirichlet series; Rankin-

Selberg method; zeros; Riemann zeta-function; Dirichlet L-functions; prime number the-

orem; arithmetic progressions; sieve method; Epstein’s zeta functions; Chowla-Selberg

formula; harmonic analysis; discontinuous groups; Selberg Trace Formula; Selberg zeta-

function; Linnik-Selberg conjecture; smallest positive eigenvalue; Laplacian; automor-

phic forms; analytic continuation; Eisenstein series; Roelcke-Selberg conjecture; mock

theta functions

Classification :

∗11-03 Historical (number theory)

11-02 Research monographs (number theory)

43-03 Historical (abstract harmonic analysis)

22-03 Historical (topological groups)

01A75 Collected or selected works

11E45 Analytic theory of forms

22E40 Discrete subgroups of Lie groups

11Fxx Discontinuous groups and automorphic forms

11Mxx Analytic theory of zeta and L-functions

Cited in ...

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