• In this book, unless indicated otherwise, by “number” we mean a “positive
• 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
n, we mean that
• When we write
f(p), we mean the infinite sum f(2) + f(3) + f(5) + f(7) +
. . . + f(p) + . . .. Similarly we write
f(p) to indicate that the summation
runs over all primes p ≤ x.
• The expressions
f(p) are analogue to the ones mentioned just
above, except that this time they stand for products and not summations.
f(d), we mean that the summation runs on all divisors d of n; by
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
• We denote by γ the Euler constant, which is defined by
γ = lim
− 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.