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If you plot the following function $$f(n) = \frac{(p_1+\ldots+p_{n})}{n} - \frac{(p_1+\ldots+p_{n-1})}{n-1}$$ you get a graph that is similar to $$f(x) = \frac{5}{4}\log(x) + \frac{1}{2}$$

From this we can state $$\frac{(p_1+\ldots+p_{n})}{n} - \frac{(p_1+\ldots+p_{n-1})}{n-1} \sim \frac{5}{4}\log(n) + \frac{1}{2}$$ $$(p_1+\ldots+p_{n}) - (p_1+\ldots+p_{n-1}) \sim \frac{5}{4}n\log(n) + \frac{1}{2}n$$ $$p_n \sim \frac{5}{4}n\log(n) + \frac{1}{2}n + \frac{(p_1+\ldots+p_{n-1})}{n-1}$$

Looking at the graph for $$\left|\frac{5}{4}n\log(n) + \frac{1}{2}n + \frac{(p_1+\ldots+p_{n-1})}{n-1} - p_n\right|$$ the maximum error seems to be $\log(n^2)^2$, which would be quite good for larger $n$'s (if it is indeed true)

Marijn
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  • You are overestimating, $p_n = n\log n + n\log \log n - n + o(n)$. – Daniel Fischer Sep 03 '15 at 12:33
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    Please show us your graphs, or at least tell us the range of $n$ they span. It's possible your approximation is reasonably good for $n$ "small," but as has been pointed out, it overestimates $p_n$ for large $n$. (A minor side remark: The running average of the primes is an increasing function of $n$, so there is no need for the absolute value sign in your formula defining $f(n)$.) – Barry Cipra Sep 03 '15 at 18:08

1 Answers1

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First of all, notation: $f(x)\sim g(x)$ means that $$ \lim_{x\to\infty}\frac{f(x)}{g(x)}=1, $$ and $f(x)\approx g(x)$ means that the two functions are 'approximately' equal (which doesn't have a hard and fast definition).

Lower-order terms disappear under $\sim,$ so asking if $p_n\sim \frac54 n\log n+\frac12n+\frac{p_1+\cdots+p_{n-1}}{n-1}$ is the same as asking if $p_n\sim \frac54 n\log n$ (as long as you already know that the latter terms are asymptotically smaller). As it happens this is false: $p_n\sim n\log n,$ and so for large enough $n$ your estimate will be about 25% too large.

Charles
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  • Thanks for the clarification. I thought I could use the $\sim$ symbol as 'roughly approximates', but perhaps I could have better used the $\approx$ symbol, as you mentioned. – Marijn Sep 06 '15 at 21:13