Consider the first Chebyshev function $\vartheta(x)=\sum_{p\le x} \log(p)$ where the sum runs over the primes less than or equal to $x$.
I wanted to approximate $\vartheta(x).$
My attempt was $f(x)=\sum_{n \ge 2}^x n^\frac{-1}{n}.$ It overcounts by about $2$ at $x=101$ giving a value of $90.177$ whereas $\vartheta(x)$ gives $88.344.$ I'm not sure how $f(x)$ performs as $x$ increases.
Is $f(x)\sim \vartheta(x)?$