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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 19. Выпуск 3.
УДК 511.3 DOI 10.22405/2226-8383-2018-19-3-35-39
К вопросу о теореме Бредихина и Линника
Джон Фридландер1 — профессор кафедры математики, Университет Торонто. e-mail: frdlndr@math. toronto. edu
Хенрик Иванец2 — профессор кафедры математики, Ратгерский университет. e-mail: iwaniec@mMh.rutgers.edu
Аннотация
Мы приводим новое доказательство теоремы Б. М. Бредихина, которая изначально была доказана путем адаптации решения проблемы Харди-Литтлвуда, полученного Лин-ником с помощью его дисперсионного метода.
Ключевые слова: простые числа, дисперсия, теорема Бомбьери-Виноградова.
Библиография: 5 названий.
Для цитирования:
Д. Фридландер, X. Иванец. К вопросу о теореме Бредихина и Линника // Чебышевский сборник, 2018, т. 19, вып. 3, с. 35-39.
CHEBYSHEVSKII SBORNIK Vol. 19. No. 3.
UDC 511.3 DOI 10.22405/2226-8383-2018-19-3-35-39
On a theorem of Bredihin and Linnik
John Friedlander — Professor of Department of Mathematics, University of Toronto. e-mail: frdlndr@math. toronto. edu
Henryk Iwaniec — Professor of Department of Mathematics, Rutgers University. e-mail: iwaniec@mMh.rutgers.edu,
Abstract
We give a new proof of a theorem of В. M. Bredihin which was originally proved by extending Linnik's solution, via his dispersion method, of a problem of Hardy and Littlewood.
Keywords: primes, dispersion, Bombieri-Vinogradov theorem.
Bibliography: 5 titles.
For citation:
J. Friedlander, H. Iwaniec, 2018, "On a theorem of Bredihin and Linnik" , Chebyshevskii sbornik, vol. 19, no. 3, pp. 35-39.
1 Supported in part by NSERC grant A5123.
2 Supported in part by NSF grant DMS-1406981
Dedicated to the memory of Yu. V. Linnik.
1. Introduction
Among the many beautiful consequences of Linnik's dispersion method is an asymptotic formula for the number of solutions to the equation
p = a2 + b2 + 1
in primes p ^ x and integers a and b. This result of 1965, due to Bredihin [2] was a follow-up to Linnik's celebrated work on the Hardy-Littlewood problem, cf. Chapter 7 of [5]. The involved arguments are lengthy and complicated, though very inventive. Due to much progress over the intervening years, much shorter arguments can now be put forward. This of course does not mean that they are shorter ab-initio. Our purpose here is to illustrate how these arguments can be applied.
Theorem 1. Let S(x) be the number of solutions to
p = a2 + b2 + 1 (1)
in integers a and b and primes p = 3( mod 8), p ^ x. We have
s«) = ^+«M10^)2)' <2>
where the constant c is given by
- = ln(i + <3'
with x being the Dirichlet character of conductor 4.
8
we do not treat them.
Note that the theorem shows that the integers p — 1 tend to have about as many representations as the sum of two squares as does a typical integer n. Recall also that, if the number of representable p — 1 is counted without multiplicitv in a and b, then the order of magnitude is given by x/(log x)3/2 by a theorem of the second-named author [4].
2. Dirichlet divisor switching
Let A = 1 * % that is
A(n) = £ x(a) (4)
ab=n
This is similiar in many respects to the divisor function t (n). The number of representations of n as the sum of two squares is equal to 4A(n). If n = 1( mod 4) then, in (4), \(a) can be replaced by x(6); therefore we can write
A(n) = £ x(a) + E *(b) (5)
a\n b\n
a^y b<n/y
for any y > 0. We can refine this partition by integrating over y against a smooth weight function. Let w(i) be a smooth function supported on 1 ^ t ^ 2 such that
¿-1
/ w(t)t-1dt = 1. (6)
J 0
Let Y ^ 1, multiply (5) bv w(y/Y) and integrate with the measure y ldy, getting
A(n) = J>(I) + w( ^ )](E X(6)) f . (7)
b\n b<y
Note that if X < n T 2X we can choose Y = \/~X so the integration in (7) runs over the segment 2VX <y< 2VX.
3. Primes in arithmetic progressions
The key input which greatly streamlines the proof is the main result of fl] which gives asvmptotics of Bombieri-Vinogradov type for the distribution of primes in arithmetic progressions and which treats moduli of the progression which go beyond the range of that which can be sucessfullv handled even on the assumption of the Generalized Riemann Hypothesis.
We state this restricted to a range somewhat lesser than that in fl], which is however sufficient for our needs and is conveniently recorded as Theorem 2.2.1 of [3].
-) - H) 1« K^)2 <8>
qTQ
(q,a)=l
(log log X\2
«x[ b b *
X
for Q = y^(log x)A with any a = 0, A ^ 0 x ^ 3, the implied constant depending only on a and A.
We actually require a slightly modified form of (8) which follows from it in two easy steps. In the first place we have
El E i - «42 (9)
1 v(qk) V iog x J
qTQ p^X
(q,a) = \ p=a( mod q) (q,k) = \ p=£( mod k)
(log log X \ 2 «x[ ]
X
for Q = y^(log x)A with any a = 0, k ^ 1, (l, k) = 1, A ^ 0 x ^ 3, the implied constant depending only on a,k and A To this end one merely splits the indexed variables into classes modulo k, which is harmless for k fixed.
In the second step we modify (9) to a counting of primes with smooth weight.
Lemma 1. Let, f (t) be a smooth function supported on 1 T t T 2. We have
P \ 1 £( P
El E f®-f® «(1°>
(q,a) = \ p=a( mod q) (q,k) = \ p=£( mod k)
for Q = y^(log x)A with any a = 0 k ^ 1, (£,k) = 1, A ^ 0 x ^ 3; the implied constant depending only on a,k, A and f.
Proof. We write
p_
f(f) = - i~ f(t)dt. A Jp/X
Given 1 T t T 2 this implies p T tX. Applying (9) with x = tX and integrating the result over t, we derive (10). ■
4. Proof of the theorem
We have
S (*) = 4 E A( ).
p—x p=3( mod 8)
We are going to evaluate
(11)
X<p-2X p=3( mod 8)
T(X) = S(2X) -S(X) = 4 E A(^)
(12)
for every X ^ 3. Applying (7) we write
T (X )=4/ E M Y) +-(I-1)
b<y X<p-2X
p=1( mod b) p=3( mod 8)
d/y
y
where we choose Y = \/~X. Here we can replace w((p — 1)/2yY) by w(p/2yY) up to an error term 0(1/yY) which contributes to T(X) a bounded amount:
T(X) = 4 /E%(&) E MY) dY)1 V + °(1).
b<y X<p-2X
p=1( mod b) p=3( mod 8)
y2yY J
Note that the integration runs over the segment jVX < y < 2\[X. Now we can apply (9) for the first term and (10) for the second term with q = b, k = 8, I = 3, getting
T (X )=/ E $}.. £>( Y) + '"< 2yY)] ? + °(X (
b<y ' X<p^2X <
log log X ^ 2 logX
n-
IH^) n (l + ^
(13)
up to an error term 0(1/y) which contributes to T(X) at most 0(\[X / logX). Now the free
¿1 =2.
Therefore,
T (X) = 2 c 1(,(2X)-„(X)) +o(X( logogoXX )').
log X
Summing this over X = 2 nx, n = 1, 2, 3,..., we derive (2).
y
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. E. Bombieri, J.В. Friedlander and H. Iwaniec, Primes in arithmetic progressions to large moduli III // it J. Amer. Math. Soc. 2, 1989, 215-224.
2. B.M. Bredihin, Binary additive problems of indeterminate type. II. Analogue of the problem of Hardy and Littlewood // Izv. Akad. Nauk SSSR Ser. Mat' 27, 1963, 577-612.
3. J.B. Friedlander and H. Iwaniec, Opera de Cribro, // Amer. Math. Soc. Colloq. Pub. 57 AMS (Providence), 2010.
4. H. Iwaniec, Primes of the type ф(х,у) + A where ф is a quadratic form // Acta Arith. 21, 1972, 203-234.
5. Yu. V. Linnik, The Dispersion Method in Binary Additive Problems (translated from the Russian by S. Schuur) / AMS (Providence), 1963.
REFERENCES
1. E. Bombieri, J.B. Friedlander and H. Iwaniec, 1989, Primes in arithmetic progressions to large moduli III, J. Amer. Math. Soc. 2, 215-224.
2. B.M. Bredihin, 1963, Binary additive problems of indeterminate type. II. Analogue of the problem of Hardy and Littlewood, Izv. Akad. Nauk SSSR Ser. Mat. 27, 577-612.
3. J.B. Friedlander and H. Iwaniec, 2010, Opera de Cribro, Amer. Math. Soc. Colloq. Pub. 57 AMS (Providence).
4. H. Iwaniec, 1972, Primes of the type ф(х,у) + A where ф is a quadratic form, Acta Arith. 21, 203-234.
5. Yu. V. Linnik, 1963, The Dispersion Method in Binary Additive Problems (translated from the Russian by S. Schuur), AMS (Providence).
Получено 27.05.2018 Принято к печати 10.10.2018
