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. xiii
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