By Alina Carmen Cojocaru
Brief yet candy -- through some distance the easiest advent 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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Additional resources for An Introduction to Sieve Methods and Their Applications
To be precise, if we let g p − log log p Sx =# p≤x ≤ log log p then lim x→ Sx x 1 =√ 2 − e−t 2 /2 dt These results have been established in [48, 49]. 1 can, for instance, be applied to n . In , Erdös and Pomerance determined the normal order of n using the Bombieri– Vinogradov theorem. In , Murty and Saidak show that the same result can be established without this theorem and by using only the elementary sieve of Eratosthenes (to be discussed in Chapter 5). 5 Exercises 43 By invoking the theory of -adic representations (see [63, 64]), one can prove certain properties about the number of prime divisors of n .
Our strategy will be to get an upper bound estimate for the number of reducible polynomials. Let = an−1 an−2 We will think of the n-tuples an−1 polynomials a1 a0 ∈ n 0 ≤ ai < H a1 a0 as corresponding to the monic xn + an−1 xn−1 + · · · + a1 x + a0 We want to count the number of tuples of that correspond to irreducible polynomials in x . So, let consist of all primes and for each prime p, let p denote the subset of tuples corresponding to irreducible polynomials The Turán sieve 52 modulo p. Let z = z H be a positive real number to be chosen later.
This has been done in . If f is not irreducible, but has r irreducible factors, then the prime ideal theorem implies rx f p ∼ log x p≤x as x → . It will be of interest to make this effective and investigate whether such a result can be used to give an efficient ‘primality test’ or ‘irreducibility test’ for an arbitrary f ∈ x . It would also be of interest to generalize these investigations to study polynomials of several variables. This leads to the study of polynomial congruences modulo p in several variables, about which much is known thanks to the spectacular development of modern algebraic geometry (see, for example, the excellent monograph of Ireland and Rosen  for an introduction).
An Introduction to Sieve Methods and Their Applications by Alina Carmen Cojocaru