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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