12 1. Arithmetic Functions The last inequality yields asymptotic estimates for the summatory functions of log(m) and 1/m. The first of these is a weak version of Stirling’s formula. Proposition 1.7. The estimates n m=1 log(m) = n log(n) n + log(n) 2 + 1 + Rn and n m=1 1 m = log(n) + γ + 1 2n + Sn hold for all positive integers n with |Rn| 3/16 and |Sn| 3/(16n2). Proof. The partial summation formula of Proposition 1.5 gives n m=1 log(m) = n log(n) n 1 [u] du u = n log(n) n + 1 + 1 2 log(n) + n 1 S(u) du u when f(n) 1 and g(x) = log(x). The last integral is bounded by 3/16 in absolute value by Proposition 1.6. Using the partial summation formula again yields n m=1 1 m = 1 n n 1 [u] du u2 = 1 + log(n) + 1 2n 1 2 1 S(u) du u2 + n S(u) du u2 . The last term is bounded by 3/(16n2) by Proposition 1.6. The real number γ = 0.5772 . . . is known as the Euler-Mascheroni con- stant. It is unknown whether this is irrational. Note that Proposition 1.7 yields the version T(x) = x log(x) x + O(log(x)) of Stirling’s formula that is most commonly applied in analytic number theory. Proposition 1.8 (Euler-Maclaurin summation formula). If A B are integers and f a continuous function on the interval [A, B] with f piecewise continuous there, then B n=A f(n) = B A f(u) du + f(A) + f(B) 2 + B A S(u)f (u) du with S(u) = u [u] 1/2 the sawtooth function.
Previous Page Next Page