1.1. The method of Chebyshev 3 by interchanging the order of summation in the double sum. Then m≤N n≤N/m Λ(n) = log(N!) = n≤N log(n). Any mapping f : N → C from the positive integers into the complex numbers is called an arithmetic function. The functions Λ and log are examples of arithmetic functions. Every arithmetic function f has a summatory function F (x) = n≤x f(n). Thus ψ is the summatory function of Λ. The summatory function T(x) def = n≤x log(n) of log is also important. The identity m≤x ψ x m = n≤x Λ(n) x n = T(x) holds for nonnegative x since log(N!) = T(x) where N = [x]. This identity is the starting point for the method of Chebyshev. Proposition 1.1. The inequalities log(2)x − log(4x) ≤ ψ(x) ≤ 2 log(2)x + log2(x) log(2) hold for x ≥ 1. Proof. The terms in the last sum in the computation T(x) − 2 T x 2 = n≤x log(n) − 2 m≤x/2 log(m) = n≤x log(n) − 2 2m≤x log(2m) + 2 2m≤x log(2) = n≤x (−1)n−1 log(n) + 2 x 2 log(2) alternate in sign and increase in magnitude. So T(x) − 2 T x 2 − 2 x 2 log(2) ≤ log([x]) for x ≥ 1. Thus log(2)x − log(4x) ≤ T(x) − 2 T x 2 ≤ log(2)x + log(x). Substituting the expression T(x) = n≤x ψ x n

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