An Introduction to Sieve Methods and Their Applications by Alina Carmen Cojocaru

By Alina Carmen Cojocaru

Brief yet candy -- via some distance the easiest creation to the topic, which would arrange you for the firehose that's the huge Sieve and its purposes: mathematics Geometry, Random Walks and Discrete teams (Cambridge Tracts in arithmetic)

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2) where X =# 0≤ p <1 For notational convenience, we interpret Rp p as Rp . Heuristically, we usually think of p as the proportion of elements of lying in p , and of Rp as the error term in this estimation. The same interpretation can be given to p q and Rp q . 1 (The Turán sieve) We keep the above setting. Let Uz = p pP z Then S z ≤ 2 X + Uz Uz Proof For each element a ∈ such that a ∈ p . Then S z =# a∈ Rp + pP z 1 Uz Rp q 2 pqPz , let N a be the number of primes p P z N a =0 ≤ 1 Uz 2 N a −U z 2 a∈ Thus the goal is to derive an upper bound for N a −U z 2 a∈ an expression that is reminiscent of the normal order method.

Denote by n k a the number of prime divisors of n that are ≡ a mod k . Show that n k a has normal order 1 log log n k 6. Let g be a non-negative bounded function defined on the primes and define gn = gp pn An = gp p≤n p Prove that g n −A x 2 = O x log log x n≤x 7. 1, prove that there is a positive constant c such that f p≤x p 1 = log log x + c + O p log x If f x ∈ x is not irreducible, but has r distinct irreducible factors in x (hence in x ), deduce that f n has normal order r log log n. 9. Using the Bombieri–Vinogradov theorem, prove that 8.

We would like to consider the problem of determining the normal order of f n . For this purpose, we proceed as in the previous section. The details follow. First, let us observe that if y n denotes the number of primes dividing n that are ≤ y and if y = x for some 0 < < 1/2, then for n ≤ x we have n = y n + n − y n = y n +O 1 since the number of prime divisors of n greater than y is O 1 . 1) n≤x Let us denote by f p the number of solutions modulo p of the congruence f x ≡ 0 mod p . 2) The normal order method 36 so that, upon interchanging summation, we must count, for fixed p, the number of integers n ≤ x that belong to f p residue classes modulo p.

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