Table Of ContentTopics in Number Theory
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 467
Topics in
Number Theory
In Honor of B. Gordon and S. Chawla
edited by
Scott D. Ahlgren
George E. Andrews
and
KenOno
Department of Mathematics,
Pennsylvania State University,
University Park, PA, U.S.A.
KLUWER ACADEMIC PUBLISHERS
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ISBN-13:978-1-4613-7988-1 e-ISBN-13:978-1-4613-0305-3
001: 10.1007/978-1-4613-0305-3
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v
TABLE OF CONTENTS
Preface
G. Andrews .................................................................. vii
Motivating the multiplicative spectrum
A. Granville and K. Soundararajan ............................................. 1
Modular Mahler measures I
F. Rodriguez Villegas ......................................................... 17
Diophantine problems in many variables: The role of additive number
theory
T. Wooley .................................................................... 49
Equations F(x,y,(il:) = 0
S. Ahlgren .................................................................... 85
A fundamental invariant in the theory of partitions
K. Alladi .................................................................... 101
Congruence properties of values of L-functions and applications
J. Bruinier, K. James, W. Kohnen, K. Ono, C. Skinner, and V. Vatsal ....... 115
On Tschirnhaus transformations
J. Buhler and Z. Reichstein ............................" . ..................... 127
Hecke operators and the nonvanishing of L-functions
J. B. Conrey and D. W. Farmer .............................................. 143
A new minor-arcs estimate for number fields
M. Davidson ................................................................. 151
Modular functions, Maple and Andrews' tenth problem
F. Garvan ................................................................... 163
Hecke characters and formal group characters
L. Guo ....................................................................... 181
On certain Gauss periods with common decomposition field
S. Gurak ..................................................................... 193
Theorems and conjectures involving rook polynomials with only real
zeros
J. Haglund, K. Ono, and D. Wagner .......................................... 207
vi
An example of an elliptic curve with a positive density of prime quadratic
twists which have rank zero
K. Jaznes .................................................................... 223
Exponents of class groups of quadratic fields
M. R. Murty ....................................... , ......................... 229
A local-global principle for densities
B. Poonen and M. Stoll ...................................................... 241
Divisibility of the specialization map for twists of abelian varieties
J. Silverman ................................................................. 245
vii
Preface
From July 31 through August 3,1997, the Pennsylvania State University hosted
the Topics in Number Theory Conference. The conference was organized by Ken
Ono and myself. By writing the preface, I am afforded the opportunity to express
my gratitude to Ken for beng the inspiring and driving force behind the whole
conference. Without his energy, enthusiasm and skill the entire event would never
have occurred.
We are extremely grateful to the sponsors of the conference: The National Sci
ence Foundation, The Penn State Conference Center and the Penn State Depart
ment of Mathematics.
The object in this conference was to provide a variety of presentations giving
a current picture of recent, significant work in number theory. There were eight
plenary lectures:
H. Darmon (McGill University),
"Non-vanishing of L-functions and their derivatives modulo p."
A. Granville (University of Georgia),
"Mean values of multiplicative functions."
C. Pomerance (University of Georgia),
"Recent results in primality testing."
C. Skinner (Princeton University),
"Deformations of Galois representations."
R. Stanley (Massachusetts Institute of Technology),
"Some interesting hyperplane arrangements."
F. Rodriguez Villegas (Princeton University),
"Modular Mahler measures."
T. Wooley (University of Michigan),
"Diophantine problems in many variables: The role of additive number
theory."
D. Zeilberger (Temple University),
"Reverse engineering in combinatorics and number theory."
The papers in this volume provide an accurate picture of many of the topics
presented at the conference including contributions from four of the plenary lectures.
On Saturday evening, August 2, the conference banquet was held at the Nittany
Lion Inn. The evening had a two-fold purpose. The first was to honor Basil Gordon
at 65. Basil has been the Ph.D. advisor to many ofthe most active young number
theorists of today. In addition, his work has been a powerful inspiration to many,
many others, including me.
viii
Besides honoring Basil Gordon, we also paid tribute to the late Sarvadaman
Chowla. Chowla was a professor at Penn State from 1963 to 1976. He was a towering
presence in number theory, throughout his tenure at Penn State. He brought a
continuous stream of famous mathematical visitors. In addition he inspired and
directed many graduate students and young assistant professors (me included) with
his exciting and eccentric "Chowla Seminar." It was only fitting that evening that
Penn State honor his memory by announcing the creation of the S. Chowla Assistant
Professorship and by also recognizing Kevin James (one of the contributors to this
volume) as the second S. Chowla Assistant Professor. The plaque honoring S.
Chowla Assistant Professorship and a framed photo of Chowla were unveiled in the
presence of his daughter Paromita Chowla, who recently retired from Penn State.
It is then with great pleasure that I commend to you these proceedings of the
Topics in Number Theory Conference.
George E. Andrews
University Park, Pennsylvania
July 25, 1998.
MOTIVATING THE MULTIPLICATIVE SPECTRUM
ANDREW GRANVILLE AND K. SOUNDARARAJAN
Dedicated to the memory of s. D. Chowla
ABSTRACT. In this article, we describe and motivate some of the results and notions
from our ongoing project [2]. The results stated here are substantially new (unless
otherwise attributed) and detailed proofs will appear in [2].
1. DEFINITIONS AND PROPERTIES OF THE SPECTRUM
Let S be a subset of the unit disc U. By F(S) we denote the class of completely
multiplicative functions f such that f(P) E S for all primes p. Our main concern
is: What numbers arise as mean-values of functions in F(S)?
Precisely, we define
L
= {~ = J~oo
rN(S) f(n) : f E F(S)} and r(S) rN(S),
n5,N
Here and henceforth, if we have a sequence of subsets IN of the unit disc U :=
{Izl ~ I}, then by writing limN-+oo IN = J we mean that z E J if and only if
there is a sequence of points ZN E IN with ZN -+ Z as N -+ 00. We call r(S) the
spectrum of the set S and our main object is to understand the spectrum. Although
we can determine the spectrum explicitly only in a few cases (see Theorem 1 below
for the most interesting of these cases), we are able to qualitatively describe it, and
obtain a lot of its geometric structure. For example, we can always determine the
boundary points of the spectrum (that is the elements ofr(S)n'll' where 'll' is the unit
circle). Another property is that the spectrum is always connected. Qualitatively,
the spectrum may be described in terms of Euler products and solutions to certain
integral equations.
We begin with a few immediate consequences of our definition:
• r(S) is a closed subset of the unit disc U.
• r(S) = r(S) (where S denotes the closure of S). Henceforth, we shall assume
that S is always closed.
• If S1 C S2 then r(S1) c r(S2)'
• r({I}) = {I}.
One of the main results of [2], which formed the initial motivation to study the
questions discussed herein, is a precise description of the spectrum of [-1,1].
The first author is a Presidential Faculty Fellow. He is also supported, in part, by the National
Science Foundation. The second author is supported by an Alfred P. Sloan dissertation fellowship
S.D. Ahlgren et al. (eds.). Topics in Number Theory. 1-15.
@ 1999 Kluwer Academic Publishers.
