Sieve Methods Lecture Notes Selberg’s Upper Bound Sieve
نویسنده
چکیده
1 Basic inequality For any real numbers ρ d satisfying ρ 1 = 1, and for any natural number m, we have d|m µ(d) d|m ρ d 2 = e|m λ e , λ e = [d 1 ,d 2 ]=e ρ d 1 ρ d 2. Our goal is to optimize the choice of (ρ d) d1. Let m = (n, P (z)), multiply by a n and sum over n using (g): say. Minimizing XG + R is quite difficult. However, if we restrict the support of ρ to d √ D, it is relatively easy to minimize G. Define (h) h(d) = p|d g(p) 1 − g(p) for d|P (z). (d 1 ,d 2) is squarefree, we have g([d 1 , d 2 ]) = g(d 1)g(d 2) g((d 1 , d 2)). To make use of this formula, however, we implicitly assume that g(p) > 0 for p ∈ P (primes with g(p) = 0 do not contribute to G, and we may simply remove them from P). Inverting (h) gives 1 g(m) = p|m 1 + 1 h(p) = d|m 1 h(d) , so that G = d 1 ,d 2 |P (z) ρ d 1 ρ d 2 g(d 1)g(d 2) d|(d 1 ,d 2) 1 h(d) = d|P (z) 1 h(d) d 1 ,d 2 |P (z) d|d 1 ,d|d 2 ρ d 1 ρ d 2 g(d 1)g(d 2) = d|P (z) 1 h(d) d|m ρ m g(m) 2 .
منابع مشابه
On the greatest prime factor of integers
Let N be a positive integer and let A and B be dense subsets of {1, . . . , N}. The purpose of this paper is to establish a good lower bound for the greatest prime factor of ab+ 1 as a and b run over the elements of A and B respectively. 1991 AMS Mathematics Subject Classification. Primary 11N30, Secondary 11L05, keywords: greatest prime factor, Selberg’s sieve, Kloosterman sums.
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