Arithmetical Aspects of the Large Sieve Inequality

199 Pages·2009·13.291 MB·English

by  Olivier Ramaré (auth.), D. S. Ramana (eds.)

Harish-Chandra Research Institute
Lecture Notes - 1
Arithmetical Aspects of the
Large Sieve Inequality
Arithmetical Aspects of the
Large Sieve Inequality
Olivier Rarnan§
CNRS, Universite Lille 1
France
With the Collaboration of
D. S. Ramana
Harish-Chandra Research Institute
Allahabad, India
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ISBN 978-81-85931-90-6 ISBN 978-93-86279-40-8 (eBook)
DOI 10.1007/978-93-86279-40-8
Preface
These lectures were given in February 2005 while I was a guest of
the Harish-Chandra Research Institute, and the bulk of these notes was
written while I was staying there.
Though this course was intended for people having some background
in analytic number theory, efforts have been made to restrict the pre
requisites to a minimum. As an effect, most of these notes can be read
with no prior knowledge in the area, except for some applications which
require the prime number Theorem for arithmetic progressions and, in
some places, the Bombieri-Vinogradov Theorem.
I wish to thank the Harish-Chandra Research Institute for giving
me the opportunity to give this series of lectures and for providing ex
tremely agreeable surroundings, the CEFIPRA programme "Analytic
and Combinatorial Number Theory", project 2801-1 directed by Profes
sors Bhowmik and Balasubramanian, for funding most of my journey,
and finally my host, Professor Adhikari, without whom none of this
would have been possible. I am also indebted to S. Baier who attended
these lectures and pointed out useful references, as well as to the other
persons in the audience for questions that helped me clarify these notes.
Professor D. Surya Ramana has been of great help during the writing
of this monograph: he has read many a new version, checked formu
lae, corrected references as well as provided a most welcomed linguistic
support. Chapter 3 is his, so is the last part of section 1.2.1 as well as
several parts of the proofs presented. Both of us would like to thank the
Indo-French Institute for Mathematics for supporting this collaboration.
O. Raman~~
Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. v
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1
1 The large sieve inequality . . . . . . . . . . . . . . . . . . . . . . . 7
1.1 Hilbertian inequalities. . . . . . . . . . . . . . . . . . . . . .. 7
1.2 The large sieve inequality . . . . . . . . . . . . . . . . . . .. 9
1.3 Introducing Farey points. . . . . . . . . . . . . . . . . . . .. 13
1.4 A digression: dual form and double large sieve. . . . . .. 14
1.5 Maximal variant . . . . . . . . . . . . . . . . . . . . . . . . .. 14
2 An extension of the classical arithmetical theory of the large sieve 17
2.1 Sequences supported on compact sets. . . . . . . . . . . .. 17
2.2 A family of arithmetical functions . . . . . . . . . . . . . .. 18
2.3 An identity. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20
2.4 The Brun-Titchmarsh Theorem. . . . . . .. 23
2.5 The Bombieri-Davenport Theorem ...... . 24
2.6 A detour towards lower bounds . . . . . . . . . . . . . . . . . 26
3 Some general remarks on arithmetical functions. 27
4 A geometrical interpretation . 31
4.1 Local couplings . . . . . . 31
4.2 The Fourier structure .. 32
4.3 Special cases ........................... . 35
4.4 Reduction to local properties . . . . . . . . . . . . . . . . . . 36
5 Further arithmetical applications . . . . . . . . . . . . . . . . . .. 39
5.1 On a large sieve extension of the Brun-Titchmarsh Theorem 39
5.2 Improving on the large sieve inequality for sifted sequences 42
5.3 An improved large sieve inequality for primes . . . . . . .. 42
5.4 A consequence for quadratic sequences. 44
5.5 A Bombieri-Vinogradov type Theorem 47
6 The Siegel zero effect . . . . . . . . . . . . . . 51
6.1 Zeros free regions and Siegel zeros. . . . . 51
6.2 Gallagher's prime number Theorem. . . . 53
6.3 Siegel zero and Brun-Titchmarsh Theorem. . . . . . . . .. 54
6.4 The Siegel zero effect . . . . . . . . . . . . . . . 56
6.5 A detour: the precursory theorem of Linnik . 57
6.6 And what about quadratic residues? . . . . . 60
viii Contents
7 A weighted hermitian inequality . . . 61
8 A first use of local models . . . . . . . . . . 63
8.1 Improving on the Brun-Titchmarsh Theorem . 63
8.2 Integers coprime to a fixed modulus in an interval. 64
8.3 Some auxiliary estimates on multiplicative functions. . .. 66
8.4 Local models for the sequence of primes . . . . . . . . . .. 69
8.5 Using the hermitian inequality. . . . . . . . . . . . . . . .. 72
8.6 Generalization to a weighted sieve bound. . . . . . . . .. 73
9 Twin primes and local models . . . . . . . . . . . . . . . . . . . . 75
9.1 The local model for twin primes . . 75
9.2 Estimation of the remainder term 77
9.3 Main proof. . . . . . . . . . . . . . . . . . . . 78
9.4 Guessing the local model. . . . . . . . . . . . . . . . . . . .. 79
9.5 Prime k-tuples . . . . . . . . . . . . 80
10 The three primes theorem. . . . . . . . . . . . . . . . . . . . . .. 83
10.1 An approximate Bessel inequality . . . . . . . . . . . . . .. 84
10.2 Some Fourier analysis to handle the size condition. . . .. 84
10.3 A general problem. . . . . . . . . . . . . . . . . . . . . . . .. 85
10.4 Asymptotic for ryt . . . . . . . . . . . . . . . . . . . . . . . .. 87
10.5 The local model. . . . . . . . . . . . . . . . . . . . . . . . .. 90
10.6 A slight digression. . . . . . . . . . . . . . . . . . . . . . . .. 93
11 The Selberg sieve. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
11.1 Position of the problem. . . . . . . . . . . . . . . . . . . . .. 95
11.2 Bordering system associated to a compact set. . . . . . .. 95
11.3 An extremal problem. . . . . . . . . . . . . . . . . . . . . .. 96
11.4 More on compact sets. . . . . . . . . . . . . . . . . . . . . .. 98
11.5 Pseudo-characters . . . . . . . . . . . . . . . . . . . . . . . .. 99
11.6 Selberg's bound through local models. . . . . . . . . . . . . 100
11. 7 Sieve weights in terms of local models. . . . . . . . . . .. 102
11.8 From the local models to the dual large sieve inequality .. 103
12 Fourier expansion of sieve weights . . . . . . . . . . . . . . . . .. 105
12.1 Dimension of the sieve . . . . . . . . . . . . . . . . . . . . .. 105
12.2 The Fourier coefficients. . . . . . . . . . . . . . . . . . . . . . 106
12.3 Distribution of 13K in arithmetic progressions 108
12.4 Fourier expansion of 13K . . . . . . . . . . . . . . . . . . . .. 108
13 The Selberg sieve for sequences. . . . . . . . . . . . . . . . . . .. 111
13.1 A general expression ........................ 111
13.2 The case of host sequences supported by a compact set. . 112
13.3 On a problem of Gallagher. . . . . . . . . . . . . . . . . . . . 112
Contents IX
13.4 On a problem of Gallagher, II .................. 115
13.5 On a subset of prime twins . . . . . . . . . . . . . . . . . .. 115
14 An overview ............... . 119
15 Some weighted sequences. . . . . . . . . . . . . . 121
15.1 Some special entire functions. . . . . . . . . . . . . . . . .. 121
15.2 Majorants for the characteristic function of an interval 123
15.3 A generalized large sieve inequality. . . . . . . . . . . . . .. 124
15.4 An application . . . . . . . . . . . . . . . . . . . . . . . . . .. 126
15.5 Perfect coupling . . . . . . . . . . . . . . . . . . . . . . . . .. 127
16 Small gaps between primes ....................... 129
16.1 Introduction ............................. 129
16.2 Some preliminary material. . . . . . . . . . . . . . . . . . . . 130
16.3 The actors and their local approximations. . . . . . . . .. 131
16.4 Computation of ::;ome hermitian products. . . . . . . . . . . 134
16.5 Final argument. . . . . . . . . . . . . . . . . . . . . . . . . . . 136
17 Approximating by a local model . . . . . . . . . . . . . . . . . .. 139
18 Selecting other sets of moduli ... . 143
18.1 Sieving by squares ..... . 143
18.2 A warning ............ . 144
19 Sums of two squarefree numbers . . . . . . . . . . . . . . . . . .. 147
19.1 Sketch of the proof. . . . . . . . . . . . . . . . . . . . . . .. 147
19.2 General computations ....................... 148
19.3 The hermitian product . . . . . . . . . . . . . . . . . . . . .. 150
19.4 Removing the Mq's . . . . . . . . . . . . . . . . . . . . . . .. 152
19.5 Approximating f and g. . . . . . . . . . . . . . . . . . . . . . 153
19.6 Crossed products .......................... 155
19.7 Main proof. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 156
19.8 Afterthoughts. . . . . . . . . . . . . . . . . . 156
19.9 Adding a prime and a squarefree number ........... 157
20 On a large sieve equality . . . . . . . . . . . . . . . . . . . . . . .. 159
20.1 Informal presentation. . . . . . . . . . . . . . . . . . . . . . . 159
20.2 A detour towards limit periodicity. . . . . . . . . . . . . . . 160
20.3 A large sieve equality: a pedestrian approach. . . . . . .. 163
20.4 An application . . . . . . . . . . . . . . . . . . . . . . . . . .. 166
20.5 A large sieve equality: using more advanced technology. . 171
20.6 Equality in the large sieve inequality, II ............ 174
20.7 The large sieve inequality reversed. . . . . . . . . . . . . . . 176
21 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 177
x Contents
21.1 A general mean value estimate ...... . 177
21.2 A first consequence ............. . 180
21.3 Some classical sieve bounds ........ . 181
21.4 Products of four special primes in arithmetic progressions 183
Notations .................................... 187
References . 189
Index .... 199
Introduction
The idea of the large sieve appeared for the first time in the foun
dational paper of (Linnik, 1941). Later (R€myi, 1950), (Barban, 1964),
(Roth, 1965), (Bombieri, 1965), (Davenport & Halberstam, 1966b) de
veloped it and in particular, two distinct parts emerged from these works:
(1) An analytic inequality for the values over a well-spaced set of
points of a trigonometric polynomial 8(a) = L:I<n<N une(na),
which, in arithmetical situations, most often reduces to
(0.1)
q<50Q amod*q n
for some D. depending on the length N of the trigonometric
polynomial and on Q. The best value in a general context is
D. = N - 1 + Q2 obtained independently in (Selberg, 1972) and
in (Montgomery & Vaughan, 1973).
(2) An arithmetical interpretation for L:amod*q 1 8( a/ q) 12, where this
time, information on the distribution of (Un) modulo q is intro
duced. The most popular approach goes through a lower bound
and is due to Montgomery, leading to what is sometimes referred
to as Montgomery's sieve, by reference to (Montgomery, 1968).
Today the terminology large sieve refers to a combination of the two
aforementioned steps. We refer the reader to the excellent lecture notes
(Montgomery, 1971) and the survey paper (Montgomery, 1978) for the
early part of the development, but cite here the papers of (Bombieri &
Davenport, 1968) and (Bombieri, 1971).
Almost simultaneously, (Selberg, 1949) introduced another way of
sieving, which we now describe rapidly in the following simple form for
the primes: to find an upper bound for the number of primes in the
interval ] VN, N], consider the following inequality
(0.2)
valid for any Ad'S subject to Al = 1 and Ad = 0 if d > z for some
VN.
parameter z ::; This leads to the determination of the minimum of
the quadratic form on the R.H.S. of (0.2), a method for which Selberg
designed an appropriate elementary method.