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By Alina Carmen Cojocaru

Brief yet candy -- by way of some distance the simplest advent to the topic, which would organize 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 The larger sieve Let be a (non-empty) finite set of integers and let powers. 1 (Gallagher’s larger sieve) We keep the above setting and let X = max b b∈ If t∈ t − log 2X > 0 ut then t − log 2X # ≤ t∈ t∈ where be a set of prime t − log 2X ut · is the von Mangoldt function. Some elementary sieves 18 Proof Let t ∈ and for each residue class r mod t define Z t r =# b∈ b ≡ r mod t Then = # Z t r r mod t By the Cauchy–Schwarz inequality, this is 1/2 ≤u t 1/2 Z t r 2 r mod t Hence # ut 2 ≤ 1 bb∈ b b ≡r mod t r mod t ≤# + 1 b b ∈ t b−b b=b t and we sum over t ∈ We multiply this inequality by Using t = log n tn we obtain t∈ # ut 2 By cancelling the # t ≤ # t + log 2X # 2 −# t∈ and rearranging, we establish the inequality.

In fact, for some constant c > 0, K x = li x + O xe−c √ log x where li x = x 2 dt log t is the famous logarithmic integral (let us note that, upon integration by parts, li x = x/log x + O x/log2 x ). 1 (The prime ideal theorem) p = li x + O xe−c f √ log x p≤x for some c > 0. 2 The normal number of prime divisors of a polynomial 37 We can now complete our analysis of the normal order of f n . 3)) and the corollary above, = x log log x + O x fn n≤x Also, 2 = fn n≤x 2 y f n +O n≤x y fn n≤x = 2 y f n + O x log log x n≤x We find by the Chinese remainder theorem that 2 y x = fn p q≤y p=q n≤x f p pq f q +O 1 + O x log log x where the latter error term arises from terms where p = q.

5 Exercises 1. Prove that n − log log n 2 = O x log log x n≤x 2. Let y n denote the number of prime divisors of n that are less than or equal to y. Show that y n − log log y 2 = O x log log y n≤x 3. Prove that n − log log x 2 = x log log x + O x n≤x 4. Let that n denote the number of prime powers that divide n. Show n has normal order log log n. The normal order method 44 5. Fix k ∈ and let a k = 1. 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.

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