By Andrew Granville, Zeév Rudnick

Written for graduate scholars and researchers alike, this set of lectures offers a dependent advent to the idea that of equidistribution in quantity thought. this idea is of becoming value in lots of components, together with cryptography, zeros of L-functions, Heegner issues, leading quantity thought, the speculation of quadratic kinds, and the mathematics points of quantum chaos.; the quantity brings jointly prime researchers from a variety of fields, whose available shows display attention-grabbing hyperlinks among possible disparate parts.

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Hildebrand, A. (1987) On the number of prime factors of integers without large prime divisors, J. Number Theory 25, 81–106. Kac, M. (1959) Statistical independence in probability, analysis and number theory, Vol. 12 of Carus Math. , New York, Math. Assoc. America. Khan, R. (2006) On the distribution of normal numbers, preprint. Kubilius, J. (1964) Probabilistic methods in the theory of numbers, Vol. 11 of Transl. Math. , Providence, RI, Amer. Math. Soc. Kuo, W. -R. (2006) Erd˝os–Pomerance’s conjecture on the Carlitz module, to appear.

Note that ωP (a) − µP = p∈P f p (a), and so ωP (a) − µP k = f p1 ···pk (a). ,pk ∈P a∈A a∈A As in Proposition 2 , consider more generally a∈A fr (a). Suppose r = s where the qi are distinct primes and each αi ≥ 1. Set R = i=1 qi and observe that if d = (a, R) then fr (a) = fr (d). Note that αi s i=1 qi 1 = µ(e) = a∈A e|(R/d) de|n a∈A (a,R)=d h(d) d = x µ(e)Ade e|R/d 1− h(p) µ(e)rde . + p e|(R/d) 1− h(p) + p p|(R/d) Therefore fr (a) = a∈A fr (d) d|R = x 1 a∈A (a,R)=d fr (d) d|R h(d) d = G(r)x + p|(R/d) rm E(r, m), m|R µ(e)rde fr (d) d|R e|(R/d) (13) 22 ANDREW GRANVILLE AND K.

If in addition, T > m1−ε then Dm T −1/32+ε . In the special case of an ‘RSA’ modulus m = p one can get the stronger bound T −1/8+ε . (B) Double sums over general sets We now consider, again for general modulus m, the sum S a (m, t, X, Y) = em (aθ xy ) x∈X y∈Y for arbitrary a ∈ Zm , X, Y ⊆ Zt . Here, no special structure is required. Because we have a double sum we can still get results if the sets are not too thin by means of a judicious use of Cauchy’s inequality. Specifically, we may, for example, write 2 |S a (m, t, X, Y)| ≤ |X| em (aθ ) .

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