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The Prime Number Theorem says that $\lim_{n \to \infty} \frac{\pi(n)}{\mathrm{Li}(n)} = 1$, where $\mathrm{Li}(x)$ is the Logarithm integral function $\mathrm{Li}(x) = \int_2^x \frac{1}{\log(x)}dx$. It is also the case that $\lim_{n \to \infty} \frac{\pi(n)}{n/\log(n)} = 1$, as $\mathrm{Li}(x)$ and $\log(x)$ are asymptotically equivalent. However it seems that $\mathrm{Li}(n)$ is a better approximation to $\pi(n)$ (the Mathworld article states that this has been proven, I don't know in what precise sense).

There are also results for the absolute difference between $\pi(n)$ and $\mathrm{Li}(n)$; for instance $\pi(n) - \mathrm{Li}(n)$ is known to change sign infinitely often. We also know that the Riemann Hypothesis is equivalent to $$ \pi(n) - \mathrm{Li}(n) \in O \left (\sqrt{n} \log(n) \right ).$$

In addition, Riemann showed that we have $$\pi(n) = \mathrm{Li}(n) - \frac{1}{2} \mathrm{Li}\left ( \sqrt{n} \right ) - \sum_{\rho} \mathrm{Li}(x^\rho) + \text{lower order terms}$$ where $\rho$ runs over all the nontrivial zeroes of the Riemann zeta function.

Question: Is there a sense in which $\mathrm{Li}(n)$ is the best possible approximation to $\pi(n)$? Ideally, there would be some Bohr-Mollerup type theorem: $\mathrm{Li}(n)$ is uniquely characterized as being a good approximation to $\pi(n)$ which has some properties, such as analyticity and negative second derivative. Probably $\mathrm{Li}(n)$ isn't the best possible, for instance $\mathrm{Li}(n) - \frac{1}{2} \mathrm{Li}\left ( \sqrt{n} \right )$ might be better? Riemann also suggested $$\sum_{n \geqslant 1} \frac{\mu(n)}{n} \mathrm{Li} \left (x^\frac{1}{n}\right ),$$ where $\mu(n)$ is the Möbius function.

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Not about the question, but your comment in the first paragraph - $\pi(x)=x/\log x+O(x/\log^2x)$, but the error term here is not $o(x/\log^2x)$, unlike the error term in the standard form of the PNT, $\pi(x)=\Li(x)+O(xe^{-\sqrt{x}))$. This makes precise what is meant by "$Li(x)$ is a better approximation than $x/\log x$". –  Thomas Bloom Jul 4 '10 at 13:04
    
The form of the PNT in my comment should read $\pi(x)=Li(x)+O(xe^{c\sqrt{\log x}})$. –  Thomas Bloom Jul 4 '10 at 13:06
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My understanding is that Riemann's suggested improvement (mentioned at the end of your question) is wrong, and that all the higher order terms $Li(x^{1/n})$, for $n > 1$, get swamped (for very large $x$) by the error terms coming from the non-trivial zeroes of the $\zeta$-functions. This is discussed in Edwards's book (and surely many other places too, but Edwards's book is my standard reference for $\zeta$ and the prime number theorem). –  Emerton Jul 4 '10 at 15:00
    
There are well-known explicit formulas for the very-related Mangoldt function, involving zeros of the Riemann zeta function: math.ucsb.edu/~stopple/explicit.html en.wikipedia.org/wiki/Explicit_formula –  Bo Peng Jul 4 '10 at 18:47

2 Answers 2

This is an extension of Emerton's comment. From my paperback 1990 reprint of "The Distribution of Prime Numbers" by A.E. Ingham (1932). Ingham is discussing your formula with the Mobius function, pages 105-106:

"Considerable importance was attached formerly to a function suggested by Riemann as an approximation to $\pi(x)$...This function represents $\pi(x)$ with astonishing accuracy for all values of $x$ for which $\pi(x)$ has been calculated, but we now see that its superiority over $\mathrm{Li}(x)$ is illusory...and for special values of $x$ (as large as we please) the one approximation will deviate as widely as the other from the true value."

In the next paragraph he shows that RH implies "And we can see in the same way that the function $\mathrm{Li}(x) - \frac{1}{2} \mathrm{Li}(x^{\frac{1}{2}})$ is `on the average' a better approximation than $\mathrm{Li}(x)$ to $\pi(x)$; but no importance can be attached to the latter terms in Riemann's formula even by repeated averaging."

Similar material is in Edwards pages 34-36 where he discusses relative size of terms, as Emerton comments. Edwards does not bother with comparing different approximations under the assumption of RH.

EDIT: I remember clearly that the error estimate of de la Vallee Poussin (the second comment of Thomas Bloom) shows that $\mathrm{Li}(x)$ is a better approximation for $\pi(x)$ than any rational function of $x$ and $\log x.$ I am not finding a reference that uses those exact words. However, Edwards comes pretty close in section 5.4, page 87. Here he displays the usual asymptotic expansion, formula (5), and points out that $\mathrm{Li}(x)$ is eventually a better approximation for $\pi(x)$ than the asymptotic expansion stopped at a finite number of terms. I think it likely that this also establishes the claim for any rational function of $x$ and $\log x.$

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Actually, no version of the Prime Number Theorem is needed to establish that no rational function of x and log(x) can be a better approximation to $\pi(x)$ than Li(x). The last result of Chebyshev's first (and less well known) paper on prime number number theory is that no algebraic function of x and log(x) can be a better approximation than Li(x). The result is independent of the PNT and is established by means of the Euler product formula and the nature of the singularity of $\zeta(\sigma)$ at $\sigma = 1$ as a function of a real variable. –  engelbrekt Jul 4 '10 at 21:43
    
Thanks, engelbrekt. There was a brief mention of this, page 57 in "Multiplicative Number Theory" by Harold Davenport, but I was not sure of the significance. –  Will Jagy Jul 4 '10 at 22:28

Whether for a finite set $\mathcal{R}$ of roots the approximation $$ \pi(x)\approx\mathrm{Li}(x)-\frac{1}{2}\mathrm{Li}(x^{1/2})-\sum_{\rho\in\mathcal{R}}\mathrm{Li}(x^\rho) $$ is "on average" better then $\pi(x)\approx\mathrm{Li}(x)-\frac{1}{2}\mathrm{Li}(x^{1/2})$ depends on the precise notion of average. Since $x^\rho=x^\sigma e^{i\gamma\log x}$, it might appear more natural to consider $\pi(e^t)$ instead of $\pi(x)$. Things are somewhat easier for $\Psi$, and there we get under RH $$ \lim_{T\rightarrow\infty}\frac{1}{T}\int_2^T\left(\Psi(e^t)-e^t-\sum_{\rho\in\mathcal{R}}\frac{e^{t\rho}}{\rho}\right)^2\frac{dt}{e^{t/2}} = \sum_{\rho\not\in\mathcal{R}}\frac{1}{|\rho|^2} $$ (insert explicit formula, throw away terms of small order, square out, interchange integral and summation). So after the propper rescaling including some roots actually does reduce the mean error.

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I believe this is missing a power of $2$ in the integrand. –  Peter Humphries Jul 22 '13 at 15:35

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