CHAPTER 1 Panorama of Arithmetic Functions Throughout these notes we denote by Z, Q, R, C the sets of integers, rationals, real and complex numbers, respectively. The positive integers are called natural numbers. The set N = {1, 2, 3, 4, 5, . . . } of natural numbers contains the subset of prime numbers P = {2, 3, 5, 7, 11, . . . }. We will often denote a prime number by the letter p. A function f : N C is called an arithmetic function. Sometimes an arithmetic function is extended to all Z. If f has the property (1.1) f(mn) = f(m) + f(n) for all m, n relatively prime, then f is called an additive function. Moreover, if (1.1) holds for all m, n, then f is called completely additive for example, f(n) = log n is completely additive. If f has the property (1.2) f(mn) = f(m)f(n) for all m, n relatively prime, then f is called a multiplicative function. Moreover, if (1.2) holds for all m, n, then f is called completely multiplicative for example, f(n) = n−s for a fixed s C, is completely multiplicative. If f : N C has at most a polynomial growth, then we can associate with f the Dirichlet series Df(s) = 1 f(n)n−s which converges absolutely for s = σ + it with σ sufficiently large. The product of Dirichlet series is a Dirichlet series Df(s)Dg(s) = Dh(s), with h = f g defined by (1.3) h(l) = mn=l f(m)g(n) = d|l f(d)g(l/d), which is called the Dirichlet convolution. The constant function f(n) = 1 for all n N has the Dirichlet series (1.4) ζ(s) = 1 n−s 3 http://dx.doi.org/10.1090/ulect/062/01
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