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Mar
15
revised How can we show that $\pi (x+y) - \pi(y) \le \frac{1}{3} x + C$ using the sieve of eratosthenes?
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Mar
15
comment How can we show that $\pi (x+y) - \pi(y) \le \frac{1}{3} x + C$ using the sieve of eratosthenes?
Yea, exactly TonyK.
Mar
15
revised How can we show that $\pi (x+y) - \pi(y) \le \frac{1}{3} x + C$ using the sieve of eratosthenes?
deleted 114 characters in body
Mar
15
revised How can we show that $\pi (x+y) - \pi(y) \le \frac{1}{3} x + C$ using the sieve of eratosthenes?
edited title
Mar
15
accepted How can we show $\pi (2n) \ge \frac{\log \begin{pmatrix} 2n \\n \end{pmatrix} }{\log 2n}$
Mar
15
comment How can we show $\pi (2n) \ge \frac{\log \begin{pmatrix} 2n \\n \end{pmatrix} }{\log 2n}$
Thanks, anon. So $\log m = \sum_{p\le 2n} \log p ord_{p}m \le \sum_{p\le 2n} \frac{\log 2n}{\log p} \log p \le \sum_{p\le 2n} \log 2n = \pi(2n)\log 2n $ , right ?
Mar
15
accepted How can we show that $\operatorname{ord}_{p}\left(\binom{2n}n\right) \le \frac{\log 2n}{\log p}$
Mar
15
comment How can we show that $\operatorname{ord}_{p}\left(\binom{2n}n\right) \le \frac{\log 2n}{\log p}$
Thanks, Ragib Zaman.
Mar
15
comment How can we show that $\pi (x+y) - \pi(y) \le \frac{1}{3} x + C$ using the sieve of eratosthenes?
Thank you. The prime number theorem wasnt proven yet, so the only thing that comes in question is the sieve of eratosthenes (as TMM suggests, sieving n numbers in the interval of $y<n\le x+y$ but it looks like I did it wrong (I believe).
Mar
15
asked How can we show $\pi (2n) \ge \frac{\log \begin{pmatrix} 2n \\n \end{pmatrix} }{\log 2n}$