##
**Asymptotic sieve for primes.**
*(English)*
Zbl 0926.11067

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 \(A_d(x)=\sum_{n \leq x, n \equiv 0 \bmod d}a_n\), 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 \[ \sum_{d \leq D}\mu^2(d)\bigl| r_d(x) \bigr| \leq A_1(x)/\log^B(x) \] 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 \(\mathcal A\) of integers.

It had originally been the hope that one might detect primes in this way, ideally when \(\mathcal A\) 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 A. Selberg [11. Skand. Mat. Kongr., Trondheim 1949, 13-22 (1952; Zbl 0048.03101)]. In Selberg’s example the set \(\mathcal A\) consists of the integers having an even number of prime factors. This set is devoid of primes but can be shown to satisfy the assumptions made in the traditional sieve method, which accordingly cannot possibly detect primes without the injection of some suitable extra hypotheses.

There are results in the literature in which the sieve method has been used on specific problems in conjunction with additional input from analytic ideas to detect primes, notably in short intervals \([x-x^\theta,x]\). One may, for example, consult the paper of D. R. Heath-Brown and H. Iwaniec [Invent. Math. 55, 49-69 (1979; Zbl 0424.10028)] or a more recent article [Proc. Lond. Math. Soc. (III) 72, 261-280 (1996; Zbl 0853.11076)] by R. C. Baker and G. Harman.

The authors’ approach has a certain generality. It will, for example, deal with rather “thin” sets, the only requirement in this direction being \(A_1(x) \gg A\bigl(\sqrt x \bigr)\log^2 x\). They assume that the numbers \(a_n\) satisfy an additional bilinear condition \[ \sum_m\biggl| \sum_{N<n\leq 2N, mn \leq x} \gamma(n) \mu(mn) a_{mn} \biggr| \leq A_1(x)/\log^B x, \] where \(\gamma(n) = \sum_{d\mid n, d \leq C}\mu(d)\) and \(\mu\) denotes the usual Möbius function. This is required for every (relatively small) \(C\) with \(1 \leq C \leq x/D\) and for every \(N\) with \({\sqrt D}/\Delta(x) < N < {\sqrt x}/\delta(x)\), for some \(\Delta(x)\geq \delta(x) \geq 2\). In these circumstances the authors establish a prime-counting formula \[ \sum_{p \leq x} a_p \log p = H A(x)\biggl\{ 1+O\biggl( {\log \delta(x)\over \log \Delta(x)} \biggr) \biggr\}, \] where \(H\) is given by the usual infinite product \(\prod_p \bigl( 1-g(p) \bigr)\big/ \bigl( 1-p^{-1} \bigr)\).

Initially the authors’ theorem is established under the hypothesis that the numbers \(n\) for which \(a_n \neq 0\) are all squarefree, but they describe how this hypothesis can be abandoned if the bilinear form hypothesis is strengthened somewhat. With applications in mind they also show how one may proceed when \(\gamma(n)\) is redefined to be zero when \(n\) is divisible by very small primes, not exceeding \(\Delta(x)^{1/c\log\log x}\) for a certain rather large absolute constant \(c\).

As its title suggests, this paper may be regarded as providing an extension to a “non-local” context of the asymptotic sieve of E. Bombieri [Rend. Accad. Naz. XL, V Ser. 1-2, 243-269 (1976; Zbl 0422.10042)].

In an associated publication (see the following review Zbl 0926.11068) the methods of this paper are applied to a problem about the representation of primes by a certain polynomial.

In the traditional sieve method one considers a sequence of real numbers \((a_n)\), and assumes suitable knowledge of \(A_d(x)=\sum_{n \leq x, n \equiv 0 \bmod d}a_n\), 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 \[ \sum_{d \leq D}\mu^2(d)\bigl| r_d(x) \bigr| \leq A_1(x)/\log^B(x) \] 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 \(\mathcal A\) of integers.

It had originally been the hope that one might detect primes in this way, ideally when \(\mathcal A\) 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 A. Selberg [11. Skand. Mat. Kongr., Trondheim 1949, 13-22 (1952; Zbl 0048.03101)]. In Selberg’s example the set \(\mathcal A\) consists of the integers having an even number of prime factors. This set is devoid of primes but can be shown to satisfy the assumptions made in the traditional sieve method, which accordingly cannot possibly detect primes without the injection of some suitable extra hypotheses.

There are results in the literature in which the sieve method has been used on specific problems in conjunction with additional input from analytic ideas to detect primes, notably in short intervals \([x-x^\theta,x]\). One may, for example, consult the paper of D. R. Heath-Brown and H. Iwaniec [Invent. Math. 55, 49-69 (1979; Zbl 0424.10028)] or a more recent article [Proc. Lond. Math. Soc. (III) 72, 261-280 (1996; Zbl 0853.11076)] by R. C. Baker and G. Harman.

The authors’ approach has a certain generality. It will, for example, deal with rather “thin” sets, the only requirement in this direction being \(A_1(x) \gg A\bigl(\sqrt x \bigr)\log^2 x\). They assume that the numbers \(a_n\) satisfy an additional bilinear condition \[ \sum_m\biggl| \sum_{N<n\leq 2N, mn \leq x} \gamma(n) \mu(mn) a_{mn} \biggr| \leq A_1(x)/\log^B x, \] where \(\gamma(n) = \sum_{d\mid n, d \leq C}\mu(d)\) and \(\mu\) denotes the usual Möbius function. This is required for every (relatively small) \(C\) with \(1 \leq C \leq x/D\) and for every \(N\) with \({\sqrt D}/\Delta(x) < N < {\sqrt x}/\delta(x)\), for some \(\Delta(x)\geq \delta(x) \geq 2\). In these circumstances the authors establish a prime-counting formula \[ \sum_{p \leq x} a_p \log p = H A(x)\biggl\{ 1+O\biggl( {\log \delta(x)\over \log \Delta(x)} \biggr) \biggr\}, \] where \(H\) is given by the usual infinite product \(\prod_p \bigl( 1-g(p) \bigr)\big/ \bigl( 1-p^{-1} \bigr)\).

Initially the authors’ theorem is established under the hypothesis that the numbers \(n\) for which \(a_n \neq 0\) are all squarefree, but they describe how this hypothesis can be abandoned if the bilinear form hypothesis is strengthened somewhat. With applications in mind they also show how one may proceed when \(\gamma(n)\) is redefined to be zero when \(n\) is divisible by very small primes, not exceeding \(\Delta(x)^{1/c\log\log x}\) for a certain rather large absolute constant \(c\).

As its title suggests, this paper may be regarded as providing an extension to a “non-local” context of the asymptotic sieve of E. Bombieri [Rend. Accad. Naz. XL, V Ser. 1-2, 243-269 (1976; Zbl 0422.10042)].

In an associated publication (see the following review Zbl 0926.11068) the methods of this paper are applied to a problem about the representation of primes by a certain polynomial.

Reviewer: G.Greaves (Cardiff)

### MSC:

11N35 | Sieves |