This theorem was conjectured by Legendre, Gauss, Dirichlet, Riemann, and Chebyshev. It was proved by Hadamard and de la Valley Poisson in Use the Prime Number Theorem to find the number of primes and compare the results to those found in part (A). Solution Here are some identities involving ln: ln(1 + 1/n) = 1/n + O(1/n2) nx 1/n = ln x + O(1) nx ln n = x ln x x + O(ln x) nx ln n/n = ln2 x/2 + O(1).

1. This theorem was conjectured by Legendre, Gauss, Dirichlet, Riemann, and Chebyshev. It was
proved by Hadamard and de la Valley Poisson in Use the Prime Number Theorem to find the
number of primes and compare the results to those found in part (A).
Solution
Here are some identities involving ln: ln(1 + 1/n) = 1/n + O(1/n2) nx 1/n = ln x + O(1) nx ln n =
x ln x x + O(ln x) nx ln n/n = ln2 x/2 + O(1)