divisorcount1n

Divisorcount1n is the name given to the summatory divisor function, often denoted D(n). It counts the total number of positive divisors of all integers from 1 to n. If d(k) denotes the number of positive divisors of k, then D(n) = sum_{k=1}^n d(k). Equivalently, D(n) counts the number of pairs (a, b) of positive integers with ab ≤ n.

Formally, d(k) is the divisor function, multiplicative in k, with d(k) = sum_{d|k} 1. Using this, D(n) can be written as D(n) = sum_{a≤n} floor(n/a), since it counts all pairs (a, b) with ab ≤ n. D(n) is a monotonically increasing function in n. It satisfies the asymptotic formula D(n) ~ n log n + (2γ − 1)n, where γ is the Euler–Mascheroni constant. The average value of d(k) is log k + (2γ − 1) + o(1).

The error term Δ(n) in the expansion D(n) = n log n + (2γ − 1)n + Δ(n) is the focus of the Dirichlet divisor problem. The current best-known unconditional bounds give Δ(n) = O(n^{1/4+ε}), with related refinements placing the exponent around 0.315. These results reflect the difficulty of precisely counting divisors beyond the leading terms.

Computationally, D(n) can be evaluated efficiently by using the identity D(n) = sum_{a≤n} floor(n/a) and grouping terms with equal floor values, which yields algorithms operating in roughly O(√n) time. Divisorcount1n thus serves as a bridge between elementary divisor theory and analytic estimates in number theory.