2 1. Arithmetic Functions with weight 1. This gives the weighted counting function ϑ(x) def = p≤x log(p) introduced by P. L. Chebyshev. To obtain nice formulas that are easier to analyze, it is advantageous also to count prime powers pk with weight log(p). The von Mangoldt function Λ given by Λ(pk) = log(p) when the argument is a prime power, and zero otherwise, serves this purpose. The weighted counting function ψ(x) def = pk≤x log(p) = n≤x Λ(n) was also introduced by Chebyshev. Since prime powers with exponent higher than one are quite sparse, ψ(x) mainly counts primes with weight log(p). The functions ψ(x) and ϑ(x) are thus approximately equal, and both are closely related to the counting function π(x) of the primes. In particular it will turn out that the Prime Number Theorem may equally well be expressed as one of the asymptotic relations ψ(x) ∼ x or ϑ(x) ∼ x. These formulations are often more convenient. The integers n = pm in the interval 0 n ≤ N that are divisible by a prescribed prime p are given by the integer solutions m of the inequality 0 m ≤ N/p. The largest integer less than or equal to a real number x is denoted by [x]. It is called the integer part of x or the Gauss bracket. Clearly [N/p] is the number of integers n as above. The same reasoning shows that, of these integers, exactly [N/pk] are divisible by pk. This observation allows us to write down the prime factorization N! = pk p[N/pk] of the factorial, due to A.-M. Legendre. The product is taken over all prime powers, but has only finitely many factors different from 1 because [N/pk] = 0 when pk N. The importance of the identity lies in the fact that the left- hand side does not contain the primes explicitly, and is susceptible of being estimated analytically. Taking the logarithm on both sides of the Legendre identity yields n≤N Λ(n) N n = pk log(p) N pk = log(N!). Now pk log(p) N pk = pk log(p) mpk≤N 1 = m≤N pk≤N/m log(p)

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