Analytic Number Theory: Proceedings of a Conference In Honor of Heini Halberstam Volume 1

452 Pages·1996·6.777 MB·English

by  Krishnaswami Alladi (auth.), Bruce C. Berndt, Harold G. Diamond, Adolf J. Hildebrand (eds.)

Progress in Mathematics
Volume 138
Series Editors
Hyman Bass
Joseph Oesterle
Alan Weinstein
Analytic Number Theory
Volume 1
Proceedings of a Conference
In Honor of Heini Halberstam
Bruce C. Berndt
Harold G. Diamond
Adolf J. Hildebrand
Editors
Birkhauser
Boston • Basel • Berlin
Bruce C. Berndt
Harold G. Diamond
Adolf J. Hildebrand
Department of Mathematics
University of Illinois
Urbana, IL 61801
Printed on acid-free paper
i®
© 1996 Birkhauser Boston
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ISBN-13: 978-1-4612-8645-5 e-ISBN-13: 978-1-4612-4086-0
DOl: 10.107/978-1-4612-4086-0
Typesetting by the editors in 1l9'
9 8 7 6 5 432
Contents of Volume 1
Contents of Volume 2 .................................................... vii
Preface ................................................................... ix
List of Participants ....................................................... xi
Evening Programs ....................................................... xiii
K. Alladi .................................................................. l
Weighted partition identities and applications
G. E. Andrews ........................................................... 17
Rogers-Ramanujan polynomials for modulus 6
J. Bae ................................................................... 31
On subset-sum-distinct sequences
R. C. Baker and G. Harman .............................................. 39
The Brun-Titchmarsh Theorem on average
A. Balog, H. Darmon and K. Ono ....................................... 105
Congruences for Fourier coefficients of half-integral
weight modular forms and special values of L-functions
P. T. Bateman .......................................................... 129
The asymptotic formula for the number of representa
tions of an integer as a sum of five squares
P. Bleher and J. Bourgain ............................................... 141
Distribution of the error term for the number of lattice
points inside a shifted ball
N. Boston ............................................................... 155
A probabilistic generalization of the Riemann zeta func
tion
D. Bowman ............................................................. 163
A general Heine transformation and symmetric polyno
mials of Rogers
D. Bradley .............................................................. 173
A sieve auxiliary function
T. Cochrane ............................................................ 211
Bounds on complete exponential sums
J. B. Conrey ............................................................ 225
A note on the fourth power moment of the Riemann
zeta-function
H. Daboussi ............................................................. 231
Effective estimates of exponential sums over primes
vi Contents Volume 1
H. Delange .............................................................. 245
On products of multiplicative functions of absolute value
at most 1 which are composed with linear functions
H. G. Diamond, H. Halberstam, and H.-E. Richert ...................... 265
Combinatorial sieves of dimension exceeding one II
D. Eichhorn and K. Ono ................................................ 309
Congruences for partition functions
P.D. T.A. Elliott ......................................................... 323
Fractional power large sieves
P. Erdos ................................................................ 333
Some problems I presented or planned to present in my
short talk
P. Erdos, S. W. Graham, A. Ivic and C. Pomerance ...................... 337
On the number of divisors of n!
R. Evans ................................................................ 357
Generalized Lambert series
M. Filaseta ............................................................. 371
A generalization of an irreducibility theorem of 1. Schur
J. Friedlander and H. Iwaniec ........................................... 411
Bombieri's sieve
S. W. Graham .......................................................... 431
Bh sequences
Contents of Volume 2
D. R. Heath-Brown ..................................................... 451
An estimate for Heilbronn's exponential sum
M. Helm ................................................................ 465
On B3 sequences
C. Hooley ............................................................... 471
On an elementary inequality in the theory of Diophan
tine approximation
M. N. Huxley ........................................................... 487
The integer points close to a curve II
M. Jutila ................................................................ 517
The fourth moment of Riemann's zeta-function and the
additive divisor problem
N. M. Katz and Z. Zheng ............................................... 537
On the uniform distribution of Gauss sums and Jacobi
sums
M. N. Kolountzakis ..................................................... 559
A problem of Steinhaus: Can all placements of a planar
set contain exactly one lattice point?
X.-J. Li. ................................................................ 567
A note on the Riemann-Roch Theorem for function
fields
J. Liu and T. Zhan ...................................................... 571
Estimation of exponential sums over primes in short in
tervals II
L. Lucht and K. Reifenrath .............................................. 607
Weighted Wiener-Levy theorems
W. Luo ................................................................. 621
Spectral mean-values of automorphic L-functions at spe
cial points
H. Maier ................................................................ 633
The size of the coefficients of cyclotomic polynomials
Y. Motohashi ........................................................... 641
On Kuznetsov's trace formulae
M. R. Murty and C. S. Rajan .......................................... 669
Stronger multiplicity one theorems for forms of general
type on GL2
viii Contents Volume 2
M. Nair and A. Perelli .................................................. 685
A sieve fundamental lemma for polynomials in two vari-
ables
C. Pomerance ........................................................... 703
Multiplicative independence for random integers
R. A. Rankin ........................................................... 713
On certain meromorphic modular forms
E. Scourfield ............................................................ 723
Comparison of two dissimilar sums involving the largest
prime factor of an integer
H. M. Stark ............................................................. 737
On the determination of an L-function from one value
K. B. Stolarsky ......................................................... 745
An approximation to the q-analogue of n involving the
n-analogue of a golden number
R. C. Vaughan .......................................................... 755
Small values of Dirichlet L-functions at 1
D. Wan ................................................................. 767
Global zeta functions over number fields and function
fields
S. Wong ................................................................ 777
Consequences from the study of concentration functions
on shifted twin primes
T. D. Wooley ........................................................... 803
An affine slicing approach to certain paucity problems
L.-C. Zhang ............................................................ 817
Ramanujan's class invariants, Kronecker's limit formula
and modular equations (II)
W.-B. Zhang ........................................................... 839
Probabilistic number theory in additive arithmetic semi
groups I
Preface
On May 16 - 20, 1995, approximately 150 mathematicians gathered at the
Conference Center of the University of Illinois at Allerton Park for an Inter
national Conference on Analytic Number Theory. The meeting marked the
approaching official retirement of Heini Halberstam from the mathematics fac
ulty of the University of Illinois at Urbana-Champaign. Professor Halberstam
has been at the University since 1980, for 8 years as head of the Department of
Mathematics, and has been a leading researcher and teacher in number theory
for over forty years.
The program included invited one hour lectures by G. Andrews, J. Bour
gain, J. M. Deshouillers, H. Halberstam, D. R. Heath-Brown, H. Iwaniec, H.
L. Montgomery, R. Murty, C. Pomerance, and R. C. Vaughan, and almost one
hundred other talks of varying lengths. These volumes comprise contributions
from most of the principal speakers and from many of the other participants,
as well as some papers from mathematicians who were unable to attend. The
contents span a broad range of themes from contemporary number theory, with
the majority having an analytic flavor.
The conference was made possible by grants from the National Science
Foundation, the National Security Agency, the Institute for Mathematics and
Applications, the College of Liberal Arts and Sciences, and the Mathematics
Department of the University of Illinois. We are grateful to these agencies and
institutions for their support. We also extend our thanks to the director of the
Allerton Park Conference Center, Beverly Becker, and her stafIfor their help in
preparing the conference and for ensuring a pleasant stay for the participants.
Finally, we thank the staff and several of the graduate students of the UIUC
Mathematics Department for the work they did toward the conference and in
the production of these proceedings.
Urbana, Illinois B. C. B.
March, 1996 H.G.D.
A. J. H.
Heini Halberstam