THE GENERAL SIEVE-METHOD AND ITS PLACE IN PRIME NUMBER THEORY

Where is the name of the method used to find the upper bound for N?
What is the upper bound for N ( pi , p % , and p r )?
What is the number of steps that one takes to find an upper bound for N?

Transcription

1 THE GENERAL SIEVE-METHOD AND ITS PLACE IN PRIME NUMBER THEORY ATLE SELBERG Ever since Viggo Brun introduced his ingenious sieve-method, it has been a very important tool in connection with problems in the theory of primes. This is especially due to the extreme generality of the method, which makes it yield results where the finer analytic tools will not work. But it is also characteristic of the sieve method that it leads only to partial and incomplete results. Also it was a disadvantage that to obtain good results with the method required quite extensive numerical computations due to the complexity of the method. This complexity actually increased as various technical improvements were made by several mathematicians, among them Rademacher, Estermann, Ricci, and Buchstab. I shall in this lecture speak of a more general sieve-method, which I have been studying during the last years. This method includes Brim's method and the improvements of it as a special case. It leads to a clearer and simpler formulation of the main problems connected with the sieve-method, which makes it possible to get information about the limitations of the method. It should also be mentioned that it leads to better results than Brun's method. 1. The problem that the sieve method is concerned with can be formulated as follows: We have a set of integers n the total number of which we denote by N, and a certain set of primes pi, i = 1,2,, r, and we want to estimate the number of n's that are not divisible by any of the given primes pi. Often this is stated in an only apparently more general form, where we are counting the n's that are not congruent to a certain set of residues modulo each Pili we denote the number of n's that are not divisible by any pi by N (pi, Pi,, p r ), we have, if d in the following denotes the positive integers composed only by primes pi and p(d) is the Möbius-function, that (i) N( Pl,---,p r ) - E/»M) = ZMW) I. n d We now suppose that we have an approximate expression for the number of n's divisible by d, of the form 1 One may generalize this to ask for the number of n's that contain not more than k prime factors from the given set of pi, and counting them with weights depending on these prime factors. This is often advantageous if we want to prove that there exist numbers of a given type with a small number of prime factors. For instance, one can prove in this way that every large even number can be written as a sum of two positive numbers, of which one contains at most 2 the other at most 3 prime factors. 286

2 SIEVE-METHOD IN PRIME NUMBER THEORY Ei-^ + a, f(d) here f(d) is a multiplicative function, 2 and Rd is a remainder term about which e know nothing more than an upper bound for Rd. Prom (1) we shall then.t, 0 MW) ^(Pl,"-,Pr)-^ÇeW+flÇ & - y S( 1 -^) + #?'*'- here 1 g 0 ^ 1. The disadvantage of (3), however, is that, except in the al- Lost trivial case when r is very small compared to N, the remainder term will 3 much larger than the main term, so that (3) is of no use. The sieve-method is devised to take care of this difficulty. It is based upon îe following consideration, Instead of determining N(pi, p%,, p r ) directly, e shall try to find upper and lower bounds for it by replacing the expression ZP(d) occurring in (1) with a similarly built expression which respectively Lajorizes or minorizes it, and at the same time reduces the remainder term to a.asonable size. Thus, to find an upper bound for N(pi,, p r ), we take a set of real numbers i with pi = 1, and such that for any integer n, 0 Z P<* è Z /*(d); ien we get an upper bound >) N{p u, pr) â N E g^r + E I Pä I I Be. d f{d) d /"hat remains is then the problem of determining p's which satisfy (4) and make ìe right-hand side of (5) as small as possible. Similarly, if we have a set of p/s with pi = 1, and such that for every integer n 0 e get a lower bound Z Pd g Z /i(d)i r ) tf(pi,-- d j{d) d \ Pd\ Rd, id the problem is to choose the p's which make the right-hand side a maximum ider the conditions (6). The problems of finding an upper and a lower bound for N(pi,, p r ) are ms reduced to two extremal problems, which unfortunately seem to be very 2 This is the form that usually occurs in applications. One may also consider more gênai forms of the leading term in (2).

3 288 ATLE SELBERG difficult to solve. Therefore, we shall have to be satisfied, if by introducing mor< restrictions on the p's we can get a new extremal problem which is such that w< can solve it. This, of course, will imply that our result will most likely not be th< best possible. However, in some cases, we can actually obtain best possible results. 2. It would take too long to try to go into details about the principles we ma] use advantageously in order to get an extremal problem that we can solve, anc which give a good result, so that only a few hints will have to do. First, if we look at the right-hand side of (5) or (7), we see that in order tha we should have a good result, the second term Z<* I Pd I I Rd \ must not be to< large. Since, in the cases that are of most interest, the Rd are riot very large this can be obtained by restricting the size of Z<* Pd in a suitable way. Thii one may try to obtain by requiring that, except for a certain number of them the p's should be zero, for instance, by taking pd = 0 for d > z, where z is suitably chosen, we may expect that Z<* I Pd is not essentially greater than z in orde: of magnitude. 3 Then comes the problem of satisfying the inequalities (4) or (6). For (4) thii may be done in a very simple manner by taking a set of real numbers Xd witl Xi = 1 and putting (8) Pd = Z X dl Xd 2, K = (di, d 2 ) d\d%lk=*d Then Z Pd = {Z M 2 = Z J*(<0, so that (4) is satisfied. In order to make pd = 0 for d > z, we require thatx<* = ( for d > z 112. It then remains to make the first term on the right-hand side of (5 as small as possible. This term takes the form d lt d 2 g z^j[ßi)j\a2) and our problem is now simply to find the minimum of this expression under the condition Xi = 1. This can easily be done, the minimum being (10) where N MV)' <*_ *'» /'(d) fm - w 5('-s)' 3 This can actually be proved to be the case if 2d pd/f(d) is small, and the p's satisf; either (4) or (6).

4 SIEVE-METHOD IN PRIME NUMBER THEORY 289 If z then is chosen suitably, the term Z<* I Pd I I Rd becomes small enough in comparison with (10) to give us a good upper bound (5). The case of the lower bound is essentially more complicated since it is not quite so simple to satisfy the inequalities (6) as the inequalities (4). One way of doing this is, for instance, to write Pd = 2-. Xd! Xd 2, where p is the largest prime dividing d, and further take X p = 1 for all p. Then (6) is automatically satisfied. Further, if we prescribe that Xd = 0 for d > (zp) 112 where p is the largest prime dividing d, we have pd = 0 for d > z. We have then to determine the maximum of the first term on the right-hand side of (7). This can actually be done, but is considerably more complicated than in the preceding case. There are also some alternative methods, which however all involve quite extensive computations if one tries to get a good result. Finally, it should be mentioned that there are certain principles, by which in many cases one can step by step improve the results obtained for the upper or lower bound by the previously mentioned methods. Unfortunately this procedure also requires quite extensive computations in most cases. The results that one obtains by these methods are better than those obtained by the classical sieve-method, but in most cases certainly do not represent the best possible result since we have subjected our p's to rather severe restrictions in order to get an extremal-problem that we could solve. 3. As long as we cannot solve the extremal problems connected with the sievemethod in their general form, it is of interest to have bounds for these extremal values. Such bounds in one direction are of course given by the results obtained for the restricted extremal problems. But for the cases of most interest for number-theory, for instance, when the numbers n are the values taken by an integral-valued polynomial P (x) without fixed prime divisors as the argument x runs over N consecutive integers, and the set of primes pi,, p r is the set of all primes less than a certain number, we can prove interesting results in the other direction. That is, we can give bounds, which no result obtained by the sieve-method can be better than. In particular these results show that for some problems which have been attacked repeatedly by means of the sieve-method, a solution in this way is certainly not possible. They also show, what is more surprising, that in some special cases the results obtained by the methods explained in 2 are actually the best possible results. The reason that the sieve-method cannot give "too good" results can be said to be that it is not very sensitive to the order of magnitude of the remainder terms Rd in a certain sense, for instance, if we have n = P(x) as x runs over N successive integers and P(x) is an integral-valued polynomial without fixed prime divisors. In this case if u(d) denotes the number of solutions of the congruence P(x) = 0 (mod d), we have l/f(d) = u(d)/d, and for Rd we have the result

5 290 ATLE SELBERG \Rd\ _ë u(d). However, we get essentially the same results by the sieve-method, if we suppose only that 7? _ 0( <d)n \ ttd U \d(log(n/d))"j' for a sufficiently high exponent k which depends on the number of irreducible factors into which P(x) can be factored. This remark makes it possible to replace the original problem by one which is essentially equivalent with respect to the sieve-method, but for which we try to make the corresponding N(pi,, p' r ) as large or as small as possible, and thus get results of the type we want. Let us consider for instance the simple example that the n's are N consecutive integers, and we try to estimate an upper bound of the number of these integers that have nò prime factor less than = N a where 0 < a < 1. We write for brevity N( ) instead of N(pi,, p r ). In this case we have f(d) = d and ] Rd ^ 1, so that (5) takes the form dd TOs-ivE^ + ElP-l. d ci d Now the method explained in 2 gives very easily an upper bound of the form 0(N/log N), so that we may limit ourselves to consider the case when: < 12 >??- (&s>?i«i- (Hry) From this and (4) one can deduce (120 EL_ J=O(]o g i\0. d a Now if we consider the set of all positive integers n f _ä N, with an odd number of prime-factors, we have 2 1 = ^ + O fç e- (logwd» 1/2 V f 2d \d J by virtue of a well-known result from analytic number-theory. Thus if we apply our sieve on this set of numbers n r, and denote by N f ( ) the number of them with no prime factor ^, we get (13) 2iV'(0 ^ -iv E + 0(NT, ^ e- Uo * Wd))in ). d a \ d a /, However, we cannot here be sure that the remainder term is small enough in comparison to the main term, so that we cannot draw any immediate conclusions from (13). This changes if we replace the N here by a somewhat larger number, e.g., by Ni = N 1+ where e > 0 tends to zero in a suitable way as N tends to infinity, then (12) and (12') suffice to make the remainder term small enough. From this we can draw conclusions which give us a lower bound for Z<* Pd/d,

6 SIEVE-METHOD IN PRIME NUMBER THEORY 291 id thus a lower bound for the upper bound for N(0 we can obtain from (11). he result is that we cannot obtain an upper bound which is smaller than L a similar way we can prove that a lower bound obtained by the sieve-method.nnot be larger than «2W»( tì + o(*('-^)> ere N"( ) denotes the number of positive integers SN with an even number prime factors and no prime factor ;g, If we take a = 1/2; = iv 1/2, we have id lius (14) gives N"(0 = 1. 2_V - K^)*) logiv a lower bound for the upper bound. With the method described in 2 we can tually obtain the upper bound 2AT log N + o(-±-) U \\og 2 N/ y lieh thus is essentially a best possible result. (14') becomes KW)} lus the sieve-method cannot give the right order of magnitude for the number primes SN. For 1/2 < a < 1, we can even prove that the lower bound bernes negative for large N. A If we let a decrease from 1/2 to 0 we see that N f ( ) id N"( ) differ less and less. This agrees with the fact that the sieve-method Drks better for small exponents a. We cannot prove that the bounds given by (14) and (14') essentially represent e true limits for what can be obtained by the sieve-method. However, by the ethods explained in 2 one can get very close to them. Special interest is atched to the value of the exponent a, when the lower bound stops being positive id becomes negative as a increases. We may call this the sieving limit of the 4 This is really due to the fact that we required pi to be 1. If we ask only that pi ^ 1, B lower bound can never become negative, since we may take all pa 0. However, if the ver bound is positive, it will be reached with a set of p's with pi = 1.

7 292, ATLE SELBERG problem. From the previous we have that the sieving limit in this case is S1/ By the methods in 2, we can very easily show that it is also >0.465, a bour "that can be improved step by step by further computations. Whether this pr< <iedure would actually converge to 1/2 I do not know. This analysis can easily be extended to the case that the n's are the values < an integral-valued irreducible polynomial P(x) without fixed prime divisor as x runs over N successive integers, with essentially the same results. For the case of reducible polynomials we can prove similar results, but he] there is a greater gap between these results and the bounds actually obtaine by the methods of 2. A case of special interest is the polynomial composed < two irreducible factors, which includes two well-known problems which ha 1. been attacked repeatedly by means of the sieve-method, namely the problem < the infinitude of prime twins and the Goldbach problem. The first one corresponds to taking the polynomial x(x + 2) for x = 1,2, N, and = (N + 2) 1/2. If we then could get a positive lower bound, theproble] would be solved. However, we can prove that for = N a, the lower bour actually is negative for large N if a > 1/(1 + e 3/4 ), 5 thus the sieve-method cann< solve this problem. The corresponding result holds for Goldbach's problem an in general for the case of a polynomial with only two irreducible factors. Vii can also show that there exist similar limitations for the upper bounds that ca be obtained by the sieve-method, but in this case the discrepancy between thes and the results we can actually obtain by the methods of 2 is greater. We ca show, for instance, that the best upper bound that can be obtained for the numb* of prime twins S N is f or large N more than 4 times 6 larger than what is generali assumed to be the true asymptotic value. 7 By the method described in 2 w actually can obtain a bound which is 8 times too large. I am inclined to coi jecture that this in reality is the best possible. 4. In view of this it seems that the sieve-method will be of little value for tli further progress of these problems in prime number theory which it was ori^ inally designed to deal with. But it remains as an extremely general and versati tool for establishing, for instance, upper bounds, and may perhaps, when i some way combined with an analytic approach, still play an important part i the future of these problems. INSTITUTE FOR ADVANCED STUDY, PRINCETON, N. J., U. S. A. 5 This number can actually be replaced by a slightly smaller one. 6 The factor 4 may be replaced by a slightly larger one. * Namely T(N) ~ /W/log 2 N with k = 2n^ 3(1 - l/(p - l) 2 ) =