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1987 | Book

Multiplicative Functions and Small Divisors Read chapter Read first chapter

Analytic Number Theory and Diophantine Problems

Proceedings of a Conference at Oklahoma State University, 1984

Editors: A. C. Adolphson, J. B. Conrey, A. Ghosh, R. I. Yager

Publisher: Birkhäuser Boston

Book Series : Progress in Mathematics

Included in: Professional Book Archive

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About this book

A conference on Analytic Number Theory and Diophantine Problems was held from June 24 to July 3, 1984 at the Oklahoma State University in Stillwater. The conference was funded by the National Science Foundation, the College of Arts and Sciences and the Department of Mathematics at Oklahoma State University. The papers in this volume represent only a portion of the many talks given at the conference. The principal speakers were Professors E. Bombieri, P. X. Gallagher, D. Goldfeld, S. Graham, R. Greenberg, H. Halberstam, C. Hooley, H. Iwaniec, D. J. Lewis, D. W. Masser, H. L. Montgomery, A. Selberg, and R. C. Vaughan. Of these, Professors Bombieri, Goldfeld, Masser, and Vaughan gave three lectures each, while Professor Hooley gave two. Special sessions were also held and most participants gave talks of at least twenty minutes each. Prof. P. Sarnak was unable to attend but a paper based on his intended talk is included in this volume. We take this opportunity to thank all participants for their (enthusiastic) support for the conference. Judging from the response, it was deemed a success. As for this volume, I take responsibility for any typographical errors that may occur in the final print. I also apologize for the delay (which was due to the many problems incurred while retyping all the papers). A. special thanks to Dollee Walker for retyping the papers and to Prof. W. H. Jaco for his support, encouragement and hard work in bringing the idea of the conference to fruition.

Table of Contents

Frontmatter
Multiplicative Functions and Small Divisors
Abstract
Let S be a set of positive integers and g be a multiplicative function. Consider the problem of estimating the sum
$${\text{S}}\left( {{\text{x,g}}} \right) = \sum\limits_{\mathop {n \leqslant x}\limits_{n \in S} } {g\left( {\text{n}} \right).} {\text{ }}$$
(1.1)
A natural way to start is to write
$$\text{g}\left( \text{n} \right)\text{ = }\sum\limits_{\text{d}\left| \text{n} \right.} {\text{h}\left( \text{d} \right)}$$
(1.2)
and reverse the order of summation. This in turn leads to the estimation of the contribution arising from the large divisors d of n, where n S, which often presents difficulties.
K. Alladi, P. Erdös, J. D. Vaaler
Lectures on the Thue Principle
Abstract
The aim of these lectures is to give an account of results obtained from the application of Thue’s idea of comparing two rational approximations to algebraic numbers in order to show that algebraic numbers cannot be approximated too well by rational numbers. In particular we will give special attention to the problem of obtaining effective measures of irrationality, or types, for various classes of algebraic numbers.
Enrico Bombieri
Polynomials with Low Height and Prescribed Vanishing
Abstract
In a recent paper [2] we obtained an improved formulation of Siegel’s classical result([9],Bd. I,p. 213, Hilfssatz) on small solutions of systems of linear equations. Our purpose here is to illustrate the use of this new version of Siegel’s lemma in the problem of constructing a simple type of auxiliary polynomial. More precisely, let k be an algebraic number field, O k its ring of integers, α1,α2,…,αJ distinct, nonzero algebraic numbers (which are not necesarily in k), and m1,m2,…,mJ positive integers. We will be interested in determining nontrivial polynomials P(X) in 0 K [X] which have degree less than N, vanish at each αj with multiplicity at least mj and have low height. In particular, the height of such plynomials will be bounded from above by a simple function of the degrees and heights of the algebraic numbers αj and the remaining data in the problem: m1,m2,…mJ, N and the field constants associated with k.
Enrico Bombieri, Jeffrey D. Vaaler
On Irregularities of Distribution and Approximate Evaluation of Certain Functions II
Abstract
Let U =[0,1]. Suppose that g is a Lebesgue-integrable function, not necessarily bounded, in U 2, and that h is any function in U 2. Let P = P(N) be a distribution of N points in U 2 such that h(y) is finite for every y ∈ P. For x = (x1,x2) in U 2, let B(x) denote the rectangle consisting of all y = (y1,y2) in U 2 satisfying 0 < yl < x1 and 0 < y2 < x2, and write
$${\rm{Z}}\left[ {P;{\rm{h}}:B({\rm{x}})} \right] = \sum\limits_{y{\kern 1pt} \in {\kern 1pt} P{\kern 1pt} \cap B({\rm{x}})} {{\rm{h(y}}).}$$
(1)
Let μ denote the Lebesgue measure in U2, and write
$${\rm{D}}\left[ {P;{\rm{h;g}};B({\rm{x}})} \right] = {\rm{Z}}\left[ {P;{\rm{h}};B({\rm{x}})} \right] - {\rm{N}}\int\limits_{B({\rm{x}})} {{\rm{g}}({\rm{y}}){\rm{d}}\mu } .$$
(2)
W. W. L. Chen
Simple Zeros of the Zeta-Function of a Quadratic Number Field, II
Abstract
Let K be a fixed quadratic extension of Q and write ζK(s) for the Dedekind zeta-function of K, where s = σ + it. It is wellknown, and easy to prove, that the number NK(T) of zeros of ζK(s) in the region 0 < σ < 1, 0 < t ≤ T satisfies
$${{\text{N}}_{\text{K}}}\left( {\text{T}} \right)\frac{{\text{T}}}{\pi } \log T$$
as T → ∞. On the other hand, not much is known about the number of \(\text{N}_\text{K}^\text{*} \left( \text{T} \right)\) that are simple.
J. B. Conrey, A. Ghosh, S. M. Gonek
Differential Difference Equations Associated with Sieves
Abstract
Our aim in this note is to analyse the differential difference equations underlying sieves of dimension κ > 1. A heuristic version of such an analysis together with some valuable numerical information was given by Iwaniec, van de Lune and te Riele [5] (see also te Riele [7]) and what we seek to do here, in effect, is to justify the conclusions of [5]. It has been shown elsewhere (in [2]) how to construct sieves of dimension κ > 1 on the basis of such information. In this connection we acknowledge also our indebtedness to the important thesis of Rawsthorne [6].
H. Diamond, H. Halberstam, H.-E. Richert
Primes in Arithmetic Progressions and Related Topics
Abstract
This paper (talk) has a dual purpose. The first is to report without proof some of the results of recent collaborative work on a number of multiplicative topics. These topics are connected by a thread which we shall follow in the reverse order so that in fact the work in each section was to a greater or lesser extent motivated by the work in the subsequent sections.
John Friedlander
Applications of Guinand’s Formula
Abstract
The explicit formula of Weil [21] connects quite general sums over primes with corresponding sums over the critical zeros of the Riemann zeta function (or more general L-functions). In the earlier version of Guinand [8], there is on the Riemann hypothesis1) a kind of Fourier duality between the differentials of the remainder terms in the prime number theorem (suitable renormalized) and in the formula counting critical zeros of the Riemann zeta function.
P. X. Gallagher
Analytic Number Theory on GL(r,R)
Abstract
There has been much progress in recent years on some classical questions in analytic number theory. This has been due in large part to the fusion of harmonic analysis on GL(2,R) with the techniques of analytic number theory, a method inspired by A. Selberg [17]. A lot of impetus has been gained by the trace formula of Kuznetsov [11], [12], which relates Kloosterman sums with eigenfunctions of the Laplacian on GL(2,R) modulo a discrete subgroup. We cite some of the most striking applications.
Dorian Goldfeld
Pair Correlation of Zeros and Primes in Short Intervals
Abstract
In 1943, A. Selberg [15] Deduced From The Riemann Hypothesis (Rh) that
$$\int\limits_{\rm{1}}^{\rm{X}} {{{\left( {\psi \left( {\left( {{\rm{1 + }}\delta } \right){\rm{x}}} \right){\rm{ - }}\psi \left( {\rm{x}} \right){\rm{ - }}\delta {\rm{x}}} \right)}^2}{{\rm{x}}^{{\rm{ - 2}}}}{\rm{dx}} \ll \delta {{\left( {{\mathop{\rm l}\nolimits} {\rm{ogX}}} \right)}^2}}$$
(1)
for X–1 ≤ δ ≤ X–1/4, X ≥ 2. Selberg was concerned with small values of δ and the constraint δ ≤ X–1/4 was imposed more for convenience than out of necessity. For Larger δ we have the following result.
Daniel A. Goldston, Hugh L. Montgomery
One and Two Dimensional Exponential Sums
Abstract
In number theory, one often encounters sums of the form Where D is a bounded domain in R k and e(w) =e2πiw. We shall Refer to the case k = 1 as the one-dimensional case, k = 2 as the two- dimensional case, etc. Our objective here is to give an exposition of van der Corput’s method for estimating the sums in (1). The one- dimensional case is well understood. Our knowledge of the two-dimensional case is fragmentary, and dimensions higher than two are terra incognita We shall review the one-dimensional case in Section 2. In Section 3 we will give an outline of what is known and what is conjectured about the two-dimensional case.
S. W. Graham, G. Kolesnik
Non-Vanishing of Certain Values of L-Functions
Abstract
Let K be an imaginary quadratic field. The L-functions that we will consider are defined by
$$L(\chi ,{\rm{s}}){\rm{ = }}\sum\limits_a {\frac{{x\left( a \right)}}{{N{{\left( a \right)}^s}}}}$$
where the sum is over the nonzero ideals of the ring of integers OK of K. Here χ is a grossencharacter of K of type Ao. That is, χ is a complex-valued multiplicative function on the ideals of OK such that \(\chi \left( {\left( \alpha \right)} \right){\rm{ = }}{\alpha ^{\rm{n}}}{\bar \alpha ^{\rm{m}}}\) for all α OK, α = 1 (mod fχ), where n, m ∈ Z and fχ is an ideal of OK (the conductor of χ). We call (n,m) the infinity type of χ. The above series defines an analytic function for Re(s) sufficiently large which can be analytically continued to the entire complex plane and satisfies a functional equation. By translating s or applying complex conjugation, we can clearly assume that χ has infinity type (n,0) with n = nχ > 0, as we will from here on. The functional equation is then as follows.
Ralph Greenberg
On Averages of Exponential Sums over Primes
Abstract
In this paper we shall be concerned with obtaining approximations to and estimates for the sum
$${{\text{S}}_{\text{N}}}(\alpha ){\text{ = }}\sum\limits_{{\text{n}} \leqslant {\text{N}}} {{\text{e}}({\text{n}}\alpha ) \wedge {\text{(n)}}}$$
where e(x) = exp(2πix), α is real, and Λ(n) is the von Mangoldt function. Although we are unable to establish the naturally conjectured results for this sum, we shall show how the introduction of averaging — in a form likely to occur in applications — can lead to substantial improvements.
Glyn Harman
The Distribution of Ω(n) among Numbers with No Large Prime Factors
Abstract
The main result concerns the distribution of Ω(n) within
$$\text{S(x,y)} = \left\{ {\text{n}:1 \leqslant \text{n} \leqslant \text{x}\,\text{and}\,\text{p} \leqslant \text{y}\,\text{if}\,\text{p}\left| \text{n} \right.} \right\}.$$
There is an average value k0 for Ω(n), and a dispersion parameter V,such that for k not too far from k0, and for large x, y with
$$2\,\,\log \log \,\text{x}\,\text{ + }\,\text{1} \leqslant \text{log}\,\,\text{y} \leqslant (\log \text{x})^{3/4} .$$
the number of solutions n of Ω(n) = k in S(x,y) is roughly exp(-V(k-k0)2) times the number of solutions n of Ω(n) = k0 in S(x,y).
In the course of the proof, machinery is developed which permits a sharpening in the same range of previous estimates for the local behaviour of ψ(x,y) as a function of x.
Douglas Hensley
On the Size of
Abstract
In his famous Habilitationsschrift of 1854 on trigonometric series and integration theory, Riemann gave the following interesting example which shows his high ingenuity of analysis and arithmetic as well.
Takeshi Kano
Another Note on Baker’s Theorem
Abstract
Recently G. Wüstholz [5], [6] proved a theorem in transcendence which includes and greatly extends many classical results. In particular it generalizes Baker’s famous theorem [2] on linear forms in logarithms, and places it within the context of arbitrary commutative group varieties.
D. W. Masser, G. Wüstholz
Sums of Polygonal Numbers
Abstract
Let m ≥ 1. The k-th polygonal number of order m+2 is the sum of the first k terms of the arithmetic progression 1, 1+m, l+2m, l+3m,… The polygonal numbers of orders 3 and 4 are the triangular numbers and squares, respectively.
Melvyn B. Nathanson
On the Density of B2-Bases
Abstract
A sequence A of positive integers is called a Sidon sequence or a B2-sequence if the pairwise sums are all distinct. If, in addition every non-zero integer appears in the set of differences we call A a B2-basis.
Andrew D. Pollington
Statistical Properties of Eigenvalues of the Hecke Operators
Abstract
Two basic questions concerning the Ramanujan τ-function concern the size and variation of these numbers:
(i)
Ramanujan conjecture: \(\left| {\tau (p)} \right| \leqslant 2\text{p}^{11/2}\) for all primes p.
 
(ii)
“Sato-Tate” conjecture: \(\text{a}_\text{p} = \frac{{\tau (\text{p})}}{{\text{p}^{11/2} }}\) is equidistributed with respect to
$$\text{d}\mu (\text{x}) = \left\{ \begin{gathered}\frac{1}{{2\pi }}\sqrt {4 - \text{x}^2 } \text{dx}\,\,\,\,\,\,\,\text{if}\,\,\left| \text{x} \right| \leqslant 2 \hfill \\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise} \hfill \\\end{gathered} \right.$$
as p → ∞. We refer to the last as the semicircle distribution.
 
Peter Sarnak
Transcendence Theory Over Non-Local Fields
Summary
For any commutative ring R let Val(R) denote the set of all multiplicative real valuations. Let δ: Val(R) → R denote the map given by ϕ → δ(ϕ):= inf {ϕ(a): a ∈ R, a≠0)}. Here R is the field of real numbers. In the first part of the present paper we show that for δ(ϕ) > 0 the quotient field of R “is” either an algebraic extension of the field Q of rational numbers, if and only if ϕ is Archimedian, or an algebraic extension of a rational function field in arbitrarily many variables, if and only if ϕ is non-Archimedian, Local fields are contained in the class of rings (R,ϕ) with δ(ϕ) = 0.
The second part of the paper is devoted to transcendence questions over groundfields k which are quotient fields of non-Archimedian valued rings (R,ϕ) with δ(ϕ) > 0. Our results include axiomatic formulations of the methods of Schneider, Gelfond and Baker, We also derive transcendence measures for certain elements of the completion of k.
Hans-Bernd Sieburg
Backmatter
Metadata
Title
Analytic Number Theory and Diophantine Problems
Editors
A. C. Adolphson
J. B. Conrey
A. Ghosh
R. I. Yager
Copyright Year
1987
Publisher
Birkhäuser Boston
Electronic ISBN
978-1-4612-4816-3
Print ISBN
978-1-4612-9173-2
DOI
https://doi.org/10.1007/978-1-4612-4816-3