logarithmic density

For any $A\subseteq \mathbb{N}$ we denote $S(n):=\sum _{k=1}^n\frac{1}{k}$. The values

$$\overline{\delta }(A)=\underset{n\to \mathrm{\infty }}{lim\; inf}\frac{\sum _{k\in A;k\le n}\frac{1}{k}}{S(n)}\mathit{\hspace{1em}\hspace{1em}}\underset{¯}{\delta }(A)=\underset{n\to \mathrm{\infty }}{lim\; sup}\frac{\sum _{k\in A;k\le n}\frac{1}{k}}{S(n)}$$

are calledlower and upper logarithmic density of $A$. If$\overline{\delta }(A)=\underset{¯}{\delta }(A)$ we denote this value by $\delta (A)$and call it the logarithmic density of $A$.

Logarithmic density can be equivalently defined as follows: If the limit

$$\delta (A)=\underset{n\to \mathrm{\infty }}{lim}\frac{\sum _{k\in A;k\le n}\frac{1}{k}}{S(n)},$$

exists, then it is called logarithmic density of $A$.

By the well-known $\gamma =\underset{n\to \mathrm{\infty }}{lim}S(n)-\ln n$defining Euler’s constant, we can see that the denominator in theabove definitions can be replaced by $\ln n$.