# About an asymptotic behavior in number theory

Where can I read about the asymptotic behavior (with $$N$$ tending to infinity) of the sum of the fractional parts obtained from dividing $$N$$ by all prime numbers up to $$N$$ divided by the number of these numbers?

With respect Kinunen Alexander Saint-Petersburg

• In symbols, $(\pi(N))^{-1}\sum_{p\le N}\{N/p\}$ where $p$ runs through primes. Mar 18 at 22:03
• $\int_0^1 \{\frac{1}{t}\}dt = 1-\gamma$ Mar 20 at 19:10

See this paper of Naslund, specifically Proposition 3. That result, along with the prime number theorem, shows that $$\frac{1}{\pi(N)} \sum_{p\leq N} \left\{ \frac{N}{p} \right\} \sim 1-\gamma,$$ where $$\gamma$$ denotes the Euler-Mascheroni constant.
• $\int_0^1 \{\frac{1}{t}\}dt = 1-\gamma$ Mar 20 at 19:11