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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

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    $\begingroup$ In symbols, $(\pi(N))^{-1}\sum_{p\le N}\{N/p\}$ where $p$ runs through primes. $\endgroup$
    – Gerry Myerson
    Commented Mar 18, 2023 at 22:03
  • $\begingroup$ $\int_0^1 \{\frac{1}{t}\}dt = 1-\gamma$ $\endgroup$
    – mathworker21
    Commented Mar 20, 2023 at 19:10

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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.

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    $\begingroup$ $\int_0^1 \{\frac{1}{t}\}dt = 1-\gamma$ $\endgroup$
    – mathworker21
    Commented Mar 20, 2023 at 19:11

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