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Also, if G is abelian, group and conjugacy sieves are identical. The group structure suggests a natural choice of orthonormal basis B for functions on G or G , as well as natural densities ν . We start with the simpler conjugacy sieve. 1 Conjugacy sieves 33 on G , where π runs over the set of (isomorphism classes of) irreducible linear representations π : G → GL(Vπ ), form an orthonormal basis of the space of functions on G invariant under conjugation, with the inner product f, g = 1 f (y)g(y).

In turn, since 1 = log log L + O(1) L for L 2, this only implies that π(N ) N , log log N if we want to estimate the number of primes N . However, the dual sieve inequality is really a different type of statement, and it carries some useful additional flexibility: for instance, it still implies that for n 3 we have {n n | ω(n) < κ log log N } N log log N for any κ ∈ ]0, 1[, the implied constant depending only on κ. 16) is of the right order of magnitude for L = N 1/2 . 8]). 5 General comments on the large sieve inequality 25 In addition, according to the Erdös–Kac theorem, we have 1 N n N |α ω(n) − log log N √ log log N β 1 →√ 2π β e−t 2 /2 dt α as N → +∞, for any fixed α, β ∈ R.

3)), which goes well beyond even the reach of the Generalized Riemann Hypothesis, and in comparison to which the best unconditional result, the Siegel–Walfisz Theorem, 26 2 The principle of the large sieve is pitifully weak in its uniformity. On the other hand, of course, sieve is usually constrained to yield inequalities only, whereas other methods can provide asymptotic formulas, often with strong error terms. We start by discussing the saving factor H , which is not where we put the main emphasis in this book.

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A Method for Combating Random Geometric Attack on Image Watermarking

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