Asymptotic sieve for primes

John Friedlander and Henryk Iwaniec

Review from Zentralblatt MATH:

The authors introduce a new type of sieve method which is capable of establishing the existence of prime numbers in suitable sequences.

In the traditional sieve method one considers a sequence of real numbers $(a_n)$, and assumes suitable knowledge of $A_d(x)=\sum_{n \le x, n \equiv 0 \bmod d}a_n$, via a formula such as $A_d(x) = g(d)A_1(x)+r_d(x)$, in which $g$ is to be a suitably chosen multiplicative function. It is important that the remainders $r_d(x)$ are to be small in some averaged sense, such as satisfying $$\sum_{d \le D}\mu^2(d)\bigl| r_d(x) \bigr| \le A_1(x)/\log^B(x)$$ for some absolute constant $B$. In this paper this inequality is assumed for some $D=D(x)$ with ${x^{2/3}}< D(x) <x$. Frequently the numbers $a_n$ are the characteristic function of some set $\cal A$ of integers.

It had originally been the hope that one might detect primes in this way, ideally when $\cal A$ is the set of shifted primes $p-2$, or some equally attractive situation. In due course a fundamental obstacle, now usually called the parity phenomenon, appeared in an example discovered by {\it A. Selberg} [11. Skand. Mat. Kongr., Trondheim 1949, 13-22 (1952; Zbl 0926.11068) the methods of this paper are applied to a problem about the representation of primes by a certain polynomial.

Reviewed by G.Greaves

Keywords: asymptotic sieve for primes; parity; parity-sensitive; bilinear hypothesis; prime-counting formula; bilinear form hypothesis

Classification (MSC2000): 11N35

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