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Divisor (algebraic geometry)

Divisor function

Periodica Mathematica Hungarica

Let d(n) be the divisor function and denote by [t] the integral part of the real number t. In this short note, we prove that $$\sum _{n\le x} \ d\Big (\Big [\frac{x}{n}\Big ]\Big ) = x\sum _{m\ge 1}\frac{\ d(m)}{m(m+1)} + O_{\varepsilon }\big (x^{11/23+\varepsilon }\big )$$ ∑ n ≤ x d ( [ x n ] ) = x ∑ m ≥ 1 d ( m ) m ( m + 1 ) + O ε ( x 11 / 23 + ε ) for $$x\rightarrow \infty $$ x → ∞ .

On a sum involving the divisor function