English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let Q be a set of primes tha t has relative density 6 among the primes, and let 4 (x,y, Q ) be the number of positive integers 5 x tha t have no prime factor 5 y from the set Q. Standard sieve methods do not seem to give an asymptotic formula for 4 (x,y , Q ) in the case tha t 5 6 1such that 1 R(x) = 0 (7) x log For example, it follows from the prime number theorem for arithmetic progressions that if Q consists of the primes belonging to a finite union of arithmetic progressions, then (1)is satisfied for every A > 0. In the case that (1) is satisfied with 0 P + l , (uGgl (u))' = 6u6-lq2(u I ) , U > p, and A, B , and are constants that depend on S. One might expect that an asymptotic relation similar to (3) holds as well in the case that 3 1. Then for any y with 0 y that are in Q . The method is based on the "Buchstabtype" identity (Throughout this paper the letter p denotes a prime number.) Most sieve methods (see [4]) would make use of a Buchstab-type identity to estimate the sifting function C(x, x l / ~ , Q) in terms of sifting functions C(x, xl /ul , Q) with u l > u. The method of Hildebrand that we use in this paper works in the opposite direction, using estimates for smaller values of u rather than larger values. In both cases we eventually arrive at a range of values of u where it is possible to estimate the sifting function directly. In the case of 4 (x, y, Q) when Q consists of all primes, this is easy since we have 4 (x, y, Q) = ~ ( x ) ~ ( y ) if 15 u 5 2. In the case that Q has density less than one, the situation is somewhat more complicated, and in place of the prime number theorem we shall appeal to a result of Wirsing. Specifically, it follows from [lo, Satz 31 that if Q satisfies (1) with 0 1) (ii) f (u) = 1+ O(ecU),u 2 1, (iii) f l (u) = O(e-u), u > 1. PROOF.If we differentiate both sides of (4), then we find that ufl(u) = 6f (u 1) 6f (u), for u > 1. Now define u-1 g(u) = u f (u) (1 6) Iuf (t)dt -1 f (t)dt. u1 Differentiating, we find that gl(u) = 0 for u > 1. Hence g is constant, and by letting u tend to 1, we see that the constant is zero. (ii) and (iii) were proved by Iwaniec 171. SIEVING THE POSITIVE INTEGERS BY SMALL PRIMES LEMMA2 . If y 2 that $*(t) 1+ &, and 0 < E < 0.5, then = ( 1 6 ) 1 0 g y / ~ f ( t ) d t u-a p 4 Q + 0(1+ log-A y), where the implied 0 constant may depend on o. 56 D . A. GOLDSTON AND K. 53. McCURLEY PROOF.We integrate by parts and replace $(t, QC) by (1-6)t + tR( t ) to obtain

Sieving the positive integers by small primes