# Prime-Detecting Sieves. (LMS-33)

Glyn Harman
Pages: 378
https://www.jstor.org/stable/j.ctt1r2fpx

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xii)
4. Notation
(pp. xiii-xvi)
5. Chapter One Introduction
(pp. 1-24)

Many problems in number theory have the form:Prove that there exist infinitely many primes in a set$\mathcal{A}$or prove that there is a prime in each set$\mathcal{A}^{(n)}$for all large n. Examples of the first include:

The twin-prime conjecture. Here one takes$\mathcal{A}=\{p+2:p\ \text{a prime}\}$.

Primes represented by polynomials. A typical problem here is whether the quadratic$n^{2} + 1$is infinitely often prime. So one takes$\mathcal{A}=\{n^{2}+1:n\in \Bbb{Z}\}$.

Examples of the second problem include:

Goldbach’s conjecture. In this case$\mathcal{A}^{(n)}=\{2n-p:p\ \text{a prime}, p\leq n\}$.

Is there a prime between consecutive squares? For this problem we take...

6. Chapter Two The Vaughan Identity
(pp. 25-46)

The crux of Vinogradov’s adaptation of the Eratosthenes-Legendre sieve, as we explained in the last chapter, is to replace a sum over primes with double sums. There are Type I sums, where one variable has no weight attached, and Type II sums, where both variables may have a weight but neither variable is too small or too large and there is often a way of bounding such sums. In the 1970s R. C. Vaughan produced a major simplification to the study of sums over primes when he introduced the identity that now bears his name. The genesis of this formula...

7. Chapter Three The Alternative Sieve
(pp. 47-64)

As we have seen, Vaughan’s method gives anidentitymost often applied to the auxiliary sums (trigonometric or character) that arise naturally from a number-theoretic question. Its limitation is that there is no grey area between it working (giving the correct asymptotic formula) and it failing (giving nothing). In this chapter we begin to consider a method introduced by the author in [59] that overcomes this disadvantage by two devices. First, the method is applied to the arithmetical information rather than the auxiliary results. Second, we can decompose our sums over primes, taking account of the signs of the sums....

8. Chapter Four The Rosser-Iwaniec Sieve
(pp. 65-82)

In this chapter we shall give a derivation of the Rosser-Iwaniec sieve (in its one-dimensional form) that is modelled on our treatment of the alternative sieve. This may seem strange since the author’s original formulation of this sieve had its genesis in the Rosser-Iwaniec construction! However, the comparison principle used before (comparing$\mathcal{A}$with$\mathcal{B}$) enables us to give a relatively straightforward exposition of the Rosser-Iwaniec sieve in the simplest case. The comparison set$\mathcal{B}$is no longer the set of all integers in an interval, but the choice we make automatically demonstrates that the Rosser-Iwaniec sieve is optimal...

9. Chapter Five Developing the Alternative Sieve
(pp. 83-102)

In this chapter we shall give different forms of the Fundamental Theorem from Chapter 3 (first given in [62]) and consider how to reverse the roles of the variables appearing implicitly and explicitly in our sums. We shall then go on to give an improvement in our result for small values of$||\xi p+\kappa||$. Finally we shall give another application: finding integers in short intervals that have a large prime factor. This will enable us to introduce the techniques we shall require in later chapters to study primes in short intervals.

The following is an alternative version of the Fundamental...

10. Chapter Six An Upper-Bound Sieve
(pp. 103-118)

For an upper-bound sieve we can work in a similar way to Chapters 3 and 5, but we now must only discard negative sums. As the simplest example, we consider giving upper bounds to the number of primespwith$||\xi p + \kappa|| < x^{\epsilon -\theta}$. As before, we write$T = x^{\theta}, U = x^{1 -2\theta},X = x^{1/2}, z = x^{1 -3\theta}$and put

$\mathcal{A} = \{n \in \mathcal{B} : ||n\xi + \kappa|| < x^{\epsilon -\theta}\}$.

Our fundamental theorem instantly gives the upper bound

$S(\mathcal{A},X) \leq S(\mathcal{A}, z) = \lambda S(\mathcal{B}, z)(1 + o(1))$,

where$\lambda = 2x^{\epsilon-\theta}$....

11. Chapter Seven Primes in Short Intervals
(pp. 119-156)

In this section we give a quick résumé of the zero-density approach to finding primes in short intervals, which the reader might find helpful in comparison with the sieve approach taken in the rest of this chapter. By Perron’s formula (1.4.7) we have

$\sum_{x-y < n \leq x} \Lambda(n)=-\frac{1}{2\pi i}\int_{c-iT}^{c+iT}\frac{\zeta^{\prime}}{\zeta}(s)\frac{x^{s}-(x-y)^{s}}{s}ds+O\left ( \frac{x\log^{2}x}{T} \right ).\ \caption(7.1.1)$

Here$c = 1 + (\log x)-1$. We shift the line of integration to$\text{Re} \ s = -1$. The pole of$\zeta^{\prime}(s)/\zeta(s)$at$s = 1$contributes$y$to the right-hand side of (7.1.1). On the other hand, the zeros$\rho = \beta +i\gamma$...

12. Chapter Eight The Brun-Titchmarsh Theorem on Average
(pp. 157-188)

We saw in Chapter 2 that, when$(a, q) = 1$, the conjectured asymptotic formula$\pi(x;q,a)=\frac{\pi(x)}{\phi(q)}(1+o(1))\quad\quad \caption (8.1.1)$holds on average over$q$up to$x^{1/2}(\log x)^{-A}$for someA. This raises the questions of what can be proved forevery qand what can be said if$q > x^{1/2}(\log x)^{-A}$. The possible existence of Siegel zeros inhibits progress for the former question, while even knowing the Generalized Riemann Hypothesis does not help with the second. As far asupper boundsare concerned we have already stated a result sufficient for our purposes in the present chapter in the exercise in Chapter 4 (the Brun-Titchmarsh inequality)....

13. Chapter Nine Primes in Almost All Intervals
(pp. 189-200)

We now consider what happens if, instead of asking for primes inallintervals$[x - y, x]$, we only demand that there are primes inmostintervals. Our notion ofmosthere is similar to that deployed in the previous chapter. We will say that a property holds for almost all intervals$[x - y, x]$if it holds forall real$x \leq X$except on an exceptional set of measure$o(X)$. Equivalently, it will hold for all$x \in \Bbb{N}, x \leq X$with$o(X)$exceptions. The methods usually make the exceptional set of measure$O(X(\log X)^{-A})$for any$A > 0$. We show first how the exponent for this...

14. Chapter Ten Combination with the Vector Sieve
(pp. 201-230)

In this chapter we are going to introduce two major developments of the method that were incorporated in the paper [11]. The first is the combination of the alternative sieve with thevector sieve. The second is the yoking together of the method with the Hardy-Littlewood circle method (see [161] for a full introduction to this subject). Along the way we must revisit work we did in Chapters 7 and 9 on primes in short intervals, although now we need the primes to be constrained to residue classes to a small modulus.

First, before we introduce the problem where the...

15. Chapter Eleven Generalizing to Algebraic Number Fields
(pp. 231-264)

We begin in this chapter to generalize our basic method to algebraic number fields. This chapter is based on joint work with Lewis and Kumchev [68, 69]. First we consider, in an abstract setting, what is needed to make the method work. Let$\mathcal{B}$be a set we want to sieve. This implies that it is a subset of some set with a multiplicative structure. The most general situation would be a free semigroup with the “primes” as basis elements (see [143], Sections 2.5 and 2.6, for analytic number theory in this abstract setting). We will want to remove members...

16. Chapter Twelve Variations on Gaussian Primes
(pp. 265-302)

In the previous chapter we considered the distribution of Gaussian primes in terms of sectors or polar boxes. In this chapter we consider what happens if we study Gaussian primes of the form$p = u+i\upsilon$where$u$is unrestricted butvis constrained to a given set. The material we present here is based on the work of Fouvry and Iwaniec [36] who considered the case of$\upsilon$belonging to a fairly dense sequence, in particular treating the case that$\upsilon$is a rational prime, and on the work of Friedlander and Iwaniec [38] who considered the case that$\upsilon$is...

17. Chapter Thirteen Primes of the Form $x^{3} + 2y^{3}$
(pp. 303-334)

The crucial point in the work of the previous chapter was the use that could be made of factorization over$\Bbb{Z}[i]$. Heath-Brown in [82] used the same basic idea to consider the form$x^{3} + 2y^{3}$that factorizes over$\Bbb{Z}[\rho]$, where here, and henceforth,$\rho =\sqrt[3]{2}$. The details of his work are very different to those of Friedlander and Iwaniec, however. The Type I and II information must be different, of course, but Heath-Brown uses Buchstab iteration in the spirit of our sieve (working in a way similar to Theorem 5.2 here) supplemented by the upper-bound Selberg sieve much as...

18. Chapter Fourteen Epilogue
(pp. 335-336)

Archimedes has been quoted as saying, “Give me a fulcrum, a lever that is long enough, and a place to stand, and I will move the earth.” An internet search will reveal many variants of this aphorism attributed to the ancient scientist. Having finished this book the reader could now fairly adapt these words to claim, “Give me sufficiently good Type I and Type II information and I will prove the twin prime conjecture, the Goldbach conjecture, and so on.” Before we look at those problems for which a solution still seems a distant reality, let us reflect on what...

19. Appendix Auxiliary Results
(pp. 337-348)
20. Bibliography
(pp. 349-360)
21. Index
(pp. 361-362)