Notation

We denote respectively by N, Z, Q, R, and C the set of positive integers,

the set of integers, the set of rational numbers, the set of real numbers, and

the set of complex numbers. At times, we shall let R+ stand for the set of

positive real numbers.

A number ξ is said to be algebraic if it is the solution of a polynomial

equation, that is, if there exist integers k ≥ 1 and a0 = 0,a1,...,ak such that

a0ξk +a1ξk−1

+···+ak−1ξ +ak = 0. A number ξ is said to be transcendental

if it is not algebraic.

We let e stand for the naperian number and we let π stand for the ratio of

the circumference of a circle to its diameter. Both e and π are transcendental

numbers.

We write γ for the Euler constant, which is defined by

γ = lim

N→∞

N

n=1

1

n

− log N = 0.5772156649 . . ..

Most of the time, we use the letters k, , m, n, r and α, β to designate

integers and x, y to designate real numbers. The letters C and c, with

or without subscript, are usually reserved for positive constants, but not

necessarily the same at each occurrence. Similarly, the letters p and q, with

or without subscript, will normally stand for prime numbers. Unless we

indicate otherwise, the sequence {pn} stands for the increasing sequence of

prime numbers, that is, the sequence 2, 3, 5, 7, 11, 13, 17, . . . .

By x , we mean the largest integer smaller or equal to x. Tied to this

function is the fractional part of x defined by {x} = x − x .

The expression

pa

b means that a is the largest integer for which

pa

| b.

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