Notations • In this book, unless indicated otherwise, by “number” we mean a “positive integer”. • The sequence p1, p2, p3, . . . stands for the sequence of prime numbers 2, 3, 5, . . . Thus pk stands for the kth prime number. • Unless indicated otherwise, the letters p and q stand for prime numbers. • By a|b, we mean that a divides b. By a |b, we mean that a does not divide b. Given a positive integer k, by pk n, we mean that pk|n but that pk+1 | n. • When we write p f(p), we mean the infinite sum f(2) + f(3) + f(5) + f(7) + . . . + f(p) + . . .. Similarly we write p≤x f(p) to indicate that the summation runs over all primes p ≤ x. • The expressions p f(p) and p≤x f(p) are analogue to the ones mentioned just above, except that this time they stand for products and not summations. • By d|n f(d), we mean that the summation runs on all divisors d of n by p|n f(p), we mean that the summation runs over all prime factors p of n. We use the corresponding notations for the products, that is d|n f(d) and p|n f(p). • We denote by γ the Euler constant, which is defined by γ = lim N→∞ N n=1 1 n − log N = 0.5772156649 . . . . • Given an integer b ≥ 2 and a number n whose digits in base b are d1, d2, . . . , dr, we sometimes use the notation n = [d1, d2, . . . , dr]b. If the base is not men- tioned, it should be understood that we are working in base 10. xiii

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