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.

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