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Let $p$ denote a prime, and let $\{x\}$ denote the fractional part of $x$.

Suppose that the following statement is true for all non-integer real numbers $x$: $\lim_{n\to\infty}\frac{\sum_{p\leq n}^\ \frac{\ln(p)}{p}\{ px \}}{\sum_{p\leq n}^\ \frac{\ln(p)}{p}}=\frac{1}{2}.$

Does that imply that $\lim_{n\to\infty}\frac{\sum_{p\leq n}^\ 1\cdot\{ px \}}{\sum_{p\leq n}^\ 1}=\frac{1}{2}?$

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    To answer the question directly: I don't believe so, or at least not without extra assumptions. I do not think that the first statement implies the second without something stronger. Going the other way however is a standard application of summation by parts. The reason is exactly the same as why Mertens estimate is weaker than the prime number theorem. That is, the statement $\sum_{p\leq n} \frac{\log p}{p} \sim \log n$ is strictly weaker than the PNT $\sum_{p\leq n} 1 \sim \frac{n}{\log n}.$2012-12-17

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This is only a partial answer, but the conclusion can be proved (directly, without using the hypothesis) when $x$ is a rational number. Suppose $x=a/b$ in lowest terms is fixed. Then $ \sum_{p\le n} \bigg\{\!\frac{pa}b\!\bigg\} = \sum_{\substack{1\le c\le b \\ (c,b)=1}} \sum_{\substack{p\le n \\ pa\equiv c\!\pmod{\!b}}} \bigg\{\!\frac{pa}b\!\bigg\} + \sum_{p\mid b} \bigg\{\!\frac{pa}b\!\bigg\}, $ because every prime $p$ either divides $b$ or is relatively prime to $b$, in which case $pa$ is relatively prime to $b$ as well. Since $\{t/b\}$ is periodic modulo $b$, this becomes \begin{align*} \sum_{p\le n} \bigg\{\!\frac{pa}b\!\bigg\} &= \sum_{\substack{1\le c\le b \\ (c,b)=1}} \bigg\{\!\frac{c}b\!\bigg\} \sum_{\substack{p\le n \\ pa\equiv c\!\pmod{\!b}}} 1 + O(1) \\ &= \sum_{\substack{1\le c\le b \\ (c,b)=1}} \frac{c}b \pi(n;b,c) + O(1), \end{align*} and so $ \lim_{n\to\infty} \frac{\sum_{p\le n} \big\{\!\frac{pa}b\!\big\}}{\sum_{p\le n} 1} = \lim_{n\to\infty} \sum_{\substack{1\le c\le b \\ (c,b)=1}} \frac{c}b \frac{\pi(n;b,c)}{\pi(n)}. $ The prime number theorem for arithmetic progressions tells us that $\pi(n;b,c)/\pi(n)$ tends to $1/\phi(b)$ when $n$ tends to infinity (since $c$ and $b$ are relatively prime); therefore $ \lim_{n\to\infty} \frac{\sum_{p\le n} \big\{\!\frac{pa}b\!\big\}}{\sum_{p\le n} 1} = \frac1{\phi(b)} \sum_{\substack{1\le c\le b \\ (c,b)=1}} \frac{c}b; $ this can easily be seen to equal $\frac12$ by pairing $c$ with $b-c$ in the sum (which works for all $b\ge3$).

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    On the second page, the denominators don't show up as 1/(p-1) like in the normal prime sum because certain quadratic modular equations have no solutions, like p^2 is congruent to 3 mod 4, resulting in all the even powers vanishing from the original sum.2012-12-17