4 1. PANORAMA OF ARITHMETIC FUNCTIONS which is called the Riemann zeta-function. Actually ζ(s) was first introduced by L. Euler who studied the distribution of prime numbers using the infinite product formula (1.5) ζ(s) = p 1 + 1 ps + 1 p2s + · · · = p 1 − 1 ps −1 . The series (1.4) and the product (1.5) converge absolutely in the half-plane s = σ + it, σ 1. Since ζ(s) for s 1 is well approximated by the integral ∞ 1 y−s dy = 1 s − 1 as s → 1, it follows from (1.5) that (1.6) p 1 ps ∼ log 1 s − 1 , as s 1, s → 1. By the Euler product for ζ(s) it follows that 1/ζ(s) also has a Dirichlet series expansion (1.7) 1 ζ(s) = p 1 − 1 ps = ∞ 1 μ(m) ms where μ(m) is the multiplicative function defined at prime powers by (1.8) μ(p) = −1, μ(pα) = 0, if α 2. This is a fascinating function (introduced by A. F. M¨ obius in 1832) which plays a fundamental role in the theory of prime numbers. Translating the obvious formula ζ(s) · ζ(s)−1 = 1 into the language of Dirichlet convolution we obtain the δ-function (1.9) δ(n) = m|n μ(m) = 1 if n = 1 0 if n = 1. Clearly the two relations (1.10) g = 1 ∗ f, f = μ ∗ g are equivalent. This equivalence is called the M¨ obius inversion more explicitly, (1.11) g(n) = d|n f(d) ⇐⇒ f(n) = d|n μ(d)g(n/d). If f, g are multiplicative, then so are f · g, f ∗ g. If g is multiplicative, then (1.12) d|n μ(d)g(d) = p|n ( 1 − g(p) ) . In various contexts one can view the left side of (1.12) as an “exclusion-inclusion” procedure of events occurring at divisors d of n with densities g(d). Then the right side of (1.12) represents the probability that none of the events associated with prime divisors of n takes place.

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