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

are called lower and upper logarithmic density of $A$. If $\overline{\delta}(A)=\underline{\delta}(A)$ 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

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

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