My basic question is the following. Let $u\in \mathbb N$ and let $p_i=1, ..., p_s$ denote all prime numbers $\le u$. How small can be a $C$ for which the inequality $$\frac{1}{\ln(p_1)} + \cdots + \frac{1}{\ln(p_s)} \le C\cdot \frac{u}{\ln u}$$ holds for all $u\in \mathbb N$?

A similar question for all natural numbers $k$ , $2\le k\le u$, reduces to $$\sum_{k=2}^{u}\frac{1}{\ln(k)}\le \int_{2}^{u+1}\frac{dx}{\ln(x)}= Li(u+1)$$ On the other hand $Li(x)=O(\frac{x}{\ln x})$ so there appears the next question. For which, possibly small, constant $C$ the inequality $Li(x)\le C\cdot \frac{x}{\ln x}$ holds for all x > 2$? This could help to answer the basic question given above.

  • $\begingroup$ This might be in: Rosser, J. Barkley; Schoenfeld, Lowell (1962). "Approximate formulas for some functions of prime numbers". Illinois J. Math. 6: 64–94. ISSN 0019-2082. Zbl 0122.05001. Or maybe: Authors: J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $ \theta (x)$ and $ \psi (x)$, Math. Comp. 29 (1975), 243-269 Or maybe: Dusart, Pierre. "Estimates of Some Functions Over Primes without R.H." (arxiv) $\endgroup$ – Linden Mar 31 '17 at 1:36

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