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How do we show that For $x,y \ge 0$ real numbers, there exists a constant C suchthat: $$\pi(x+y)-\pi(y) \le \frac{1}{3}x+C$$ Where $\pi(.)$ denotes thes prime counting function, is true?

the hint is to sieve n with $y< n \le x+y $:

$$\pi (x+y) \le 1+ \sum _{n \le x+y} 1+1-1 - \sum_{2|n}1 - \sum_{3|n}1 + \sum_{6|n}1 + \sum_{n\le x+y} 1 = $$ $$1+ \sum _{n \le x+y} 1+1-1 - \sum_{2|n}1 - \sum_{3|n}1 + \sum_{6|n}1 + [x+y]$$

because: $1\le n = dm \le x+y \Leftrightarrow \frac{1}{d}\le m \le \frac{x+y}{y} $

so: $$\sum_{n\le x+y , d|n}1 = [\frac{x+y}{d}]$$

then that gives: $$\pi (x+y) < 1+ [x+y] - [\frac{x+y}{2}] - [\frac{x+y}{3}] + [\frac{x+y}{6}]$$

so that will give: $\pi (x+y) < \frac{x+y}{3} + 3$ but also we get : $\pi(y) < \frac{y}{3} + 3$ so for any constant $C\ge 0$ it will surely hold that:

$$\pi(x+y) - \pi(y) < \frac{x}{3} \le \frac{x}{3} + C$$

Is this correct?

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There is $ \leq 1/3 \pi(x)$ in the title, but $\leq 1/3 x$ in the post. –  dtldarek Mar 14 '12 at 23:15
    
Thanks for the correction dtldarek. –  VVV Mar 14 '12 at 23:18
    
Hint: Which numbers modulo $6$ can be prime? –  TMM Mar 14 '12 at 23:22
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What if you forget about the $C$ for a moment and bring $\Delta y$ to the LHS. With $\Delta y \to 0$ you get: $$ \lim_{\Delta y\to 0} \frac{\pi(y+\Delta y)-\pi(y)}{\Delta y}=\pi(y)'\le 1/3. $$ Now use $\pi(y)\approx \frac{y}{\log y}$ and therefore $\pi(y)'\approx \frac{\log y-1}{(\log y)^2}=\frac{1}{\log y}-\frac{1}{(\log y)^2}$ which has a global maximum of $1/4$ at $y=e^2$. See here. –  draks ... Mar 15 '12 at 14:31
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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). –  VVV Mar 15 '12 at 15:53
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2 Answers

Hint: Which numbers modulo $6$ can be prime? (Certainly not those divisible by $2$ or $3\ldots$)

So on an interval of width $x$ from $y$ to $y + x$, how many primes ($\pi(y+x) - \pi(y)$) do we expect at most on this interval?

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I put my comment to your answer in the edit of my question, thanks. –  VVV Mar 15 '12 at 10:33
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You are asking us to show that your inequality holds for any $x, y \ge 0$ and for any constant $C$, which is obviously false. Perhaps this is what you mean:

Show that there exists a constant $C$ such that for any real numbers $x, y \ge 0$, $\pi(x+y)-\pi(y) \le \frac{1}{3}x+C$.

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Yea, exactly TonyK. –  VVV Mar 15 '12 at 22:52
    
@VVV: Still not right, I'm afraid: now your statement is vacuously true. I suppose that's an improvement! –  TonyK Mar 16 '12 at 8:34
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