C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge Tracts in Mathematics 70, Cambridge University Press, Cambridge-New York-Melbourne, (1976).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that the density of S equals ) where c p equals the number of integers n [0, p 1] for which p f(n) When deg f 2, a simple sieve again shows that the guess is correct. When deg f = 3, a more complicated argument is needed (see [Hoo67] or, for an improved error term, Chapter 4 of [Hoo76]) For general f with deg f 4, it is unknown whether the heuristic conjecture is correct, but A. Granville [Gra98] showed that it follows from the abc conjecture. Recall that the abc conjecture is the statement that for any # 0, there exists a constant C = C(#) 0 such that if a, b, c are ....

C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge University Press, Cambridge, 1976, Cambridge Tracts in Mathematics, No. 70.

....y) o(#(x, y) uniformly for x # # and y 2. Theorem 1 implies this result for y , and (2) does so in the wider range y (log x) 1 3 # . That #(x, 2) o(#(x, 2) is essentially due to Fermat, but already for y = 3, the conjecture that #(x, 3) o(#(x, 3) seems di#cult. Hooley [11] has shown, under assumption of several unproved hypotheses, including the Generalised Riemann Hypothesis, that the set of integers n with 2 3 prime has density 0. It is likely the same proof would go through for primes of the form 3 1. Thus, there may be a conditional proof that #(x, ....

C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge Tracts in Mathematics, No. 70, Cambridge University Press, Cambridge-New York-Melbourne, 1976.

....independently to Hong Quan Liu [9] and Jia Chaohua [8] It is very likely that our exponent of 4 9 can be improved by using ideas from these papers, but, for reasons of brevity, we do not consider these possible improvements here. In the initial stage, we follow an argument due to Chebyshev (cf. [6], Chapter 2) We begin by observing that x c log x O(x c ) # x m#x x c log m = # x m#x x c # d m #(d) # d #(d)N(d) 9) where (10) N(d) # d m x m#x x c 1, and # is von Mangoldt s function; i.e. #(d) log p if d = p a for some prime p and 0 otherwise. We decompose the ....

....T (v) # # k,l # k # l # v m[k,l]#ev N(m[k, l] We expect that the inner sum is about x c [k, l] Accordingly, we wish to choose the # k s to minimize the bilinear form # k,l # k # l [k, l] 1 subject to the restraints # 1 =1 and # k =0ifk z. From the theory of the Selberg sieve (cf. [6], pp. 8#. it is known that this conditional minimum is # (log z) 1 . Using this choice of # k and writing # d = # [k,l] d # k # l , we see that (14) may be re written as (15) T (v) # # d z 2 # d # v d m#ev d N(md) For future reference, we note also that the minimizing choice of # k ....

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C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge University Press, Cambridge, 1976.

....(see property (P4) below) To get back on this ladder one must use ladders of other controlling primes q which reach to some currently omitted multiple of p at the top of their ladder. The total number of elements in such ladders is shown to be large using a combinatorial sieve argument (compare [Hooley 1976], pp. 4 5) a contradiction results by showing that the sandpile contains more than n elements. We begin with a preliminary lemma. Lemma 5. If a(n) is divisible by a prime p, then p # n. If p #= 2, p n. Proof. The result is true if p = 2 or n # 3, so we may assume p # 3 and n # 4. We ....

C. Hooley, Applications of Sieve Methods to the Theory of Numbers, Cambridge Tracts in Math. No. 70, Cambridge Univ. Press: Cambridge 1976.

....Y ) q . As we have seen in the proof of Lemma 3, X a#Z # q e(#a q) # 1. Applying the Weil bound and using the standard reduction from complete exponential sums to incomplete ones, we obtain X Y a#Y K (a,q) 1 e(#a q) # q 1 2 # (see Lemma 4 of Chapter 2 of [4]) and the result follows. 7 Lemma 5 The following estimate holds: X n#X (n,q) 1 #(n) XP q (ln X) O(X 1 2 q # ) where P q (ln X) #(q) 2 q 2 (ln X 2# 1) 2#(q) q X d q (d) ln d d . Proof : In what follows, P # indicates that the sum is restricted to integers ....

C. Hooley, Applications of sieve methods to the theory of numbers , Cambridge Tracts in Math. 70, Cambridge, 1976.

....to this result, the best one is that the largest prime factor exceeds x q for q 0 677 (see [BakHar95] BakHar98] and also [Ho73] 3.4. A theorem of Hooley Chebyhev proved that if P x is the largest prime factor of n x n 2 1 , then P x x A . Hooley [Ho67] see also [Ho76]) improved the previous best known result of P x x logx A 1 logloglogx by Erdos [Erd52] to P x x 11 10 using the Selberg sieve. In this section we shall outline the proof given by Hooley in [Ho76] The exponent 11 10 has since been improved to q 1 202 , where q is the ....

....prime factor of n x n 2 1 , then P x x A . Hooley [Ho67] see also [Ho76] improved the previous best known result of P x x logx A 1 logloglogx by Erdos [Erd52] to P x x 11 10 using the Selberg sieve. In this section we shall outline the proof given by Hooley in [Ho76]. The exponent 11 10 has since been improved to q 1 202 , where q is the solution to 2 q 2log 2 q 5 4 , by Deshouillers and Iwaniec [DI83] see also [Dar96] THEOREM 3.4.1 ( Ho76] The largest prime factor of n x n 2 1 exceeds x 11 10 for all large ....

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Hooley C., Applications of sieve methods to the theory of numbers, Cambridge University Press, (1976).

....Division to Powering is easily seen to provide a FOM reduction. And, of course, Powering is a special case of Iterated Multiplication. It is interesting to note that many number theorists conjecture that 2 is a generator of the multiplicative group mod p for a constant fraction of all m bit primes [16, 17]. If this conjecture is true, then most of the argument of Section 3 can be carried out using only primes of this sort. There are some minor technical modifications that need to be made to the argument in Section 3 in order to completely do away with the use of prime powers and the use of primes ....

C. Hooley. Applications of Sieve Methods to the Theory of Numbers. Cambridge Tracts in Mathematics No. 70, 1970.

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C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge Tracts in Mathematics 70, Cambridge University Press, Cambridge-New York-Melbourne, (1976).

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C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge University Press, Cambridge, 1976, Cambridge Tracts in Mathematics, No. 70.

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C. Hooley, Application of Sieve Methods to the Theory of Numbers," Cambridge Univ. Press, Cambridge, U.K., 1976.

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C. Hooley, Application of Sieve methods to the Theory of Numbers, Cambridge University Press - (1976).

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C. Hooley, Applications of Sieve Methods to the Theory of Numbers, Cambridge Univ. Press, Cambridge, 1976.

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