Let the divisor function d(n) be the number of divisors of n (including n itself). For a prime p, d(p) = 2. In general, sum_(k = 1)^n d(k) = n ln n + (2γ - 1) n + O(n^θ), where γ is the Euler-Mascheroni constant. Dirichlet originally gave θ≈1/2 (Hardy and Wright 1979, p. 264; Hardy 1999, pp. 67-68), and Hardy and Landau showed in 1916 that θ>=1/4. The following table summarizes incremental progress on the upper limit (updating Hardy 1999, p. 81).

divisor function | Gauss's circle problem