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Why is the Chebyshev function

$\theta(x) = \sum_{p<=x}\log p$

useful in the proof of the prime number theorem. Does anyone have a conceptual argument to motivate why looking at $\sum_{p<=x} \log p$ is relevant and say something random like $\sum_{p<=x}\log\log p$ is not useful or for that matter any other random function $f$ and $\sum_{p<=x} f(p)$ is not relevant.

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The rate of growth of primes suggests the Chebyshev function might be useful: – Kimball Jul 19 '11 at 1:24
Primes are a multiplicative basis, which is a reason to expect that a function on product of primes is likely to be more natural than a function on say the number of primes, or sum of primes etc. Actually I think that the most natural function is the second Chebyshev function (which is the one appearing in both the complex analytic and elementary proofs of PNT, as well as in the "explicit formula"), but it's not hard to show that the two Chebyshev functions approximate each other. – Gjergji Zaimi Jul 19 '11 at 1:35
You might be interested in what T. Tao writes in this connection in his book on blogs from year three. – Franz Lemmermeyer Jul 19 '11 at 7:05
up vote 14 down vote accepted

There are several ideas here, some mentioned in the other answers:

One: When Gauss was a boy (by the dates found on his notes he was approximately 16) he noticed that the primes appear with density $ \frac{1}{\log x}$ around $x$. Then, instead of counting primes and looking at the function $\pi (x)$, lets weight by the natural density and look at $\sum_{p\leq x} \log p$. Since we are weighting by what we think is the density, we expect it to be asymptotic to be $x$.

Two: Differentiation of Dirichlet series. If $$ A(s)=\sum_{n=1}^{\infty} a_{n} n^{-s} $$ then $$ (A(s))'=-\sum_{n=1}^{\infty} a_{n} log(n) n^{-s}$$ The $\log$ term appears naturally in the differentiation of Dirichlet series. Taking the convolution of $\log n$ with the Mobius function (that is multiplying by $\frac{1}{\zeta(s)}$) then gives the $\Lambda(n)$ mentioned above. The $\mu$ function is really the special thing here, not the logarithm.

Expanding on this, there are other weightings besides $\log p$ which arise naturally from taking derivatives. Instead we can look at $\zeta^{''}(s)=\sum_{n=1}^\infty (\log n)^2 n^{-s}$, and then multiply by $\frac{1}{\zeta(s)}$ as before. This leads us to examine the sum $$\sum_{n\leq x} (\mu*\log^2 )(n)$$ (The $*$ is Dirichlet convolution) By looking at the above sum, Selberg was able to prove his famous identity which was at the center of the first elementary proof of the prime number theorem:

$$\sum_{p \leq x} log^2 p +\sum_{pq\leq x}(\log p)(\log q) =2x\log x +O(x).$$

Three: The primes are intimately connected to the zeros of $\zeta(s)$, and contour integrals of $\frac{1}{\zeta(s)}$. (Notice it was featured everywhere here so far) We can actually prove that

$$\sum_{p^k \leq x} \log p= x - \sum_{\rho :\ \zeta(\rho)=0} \frac{x^{\rho}}{\rho} +\frac{\zeta'(0)}{\zeta(0)} $$

Notice that the above is an equality, which is remarkable since the left hand side is a step function. (Somehow, at prime powers all of the zeros of zeta conspire and make the function jump.)

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The distribution of sums $\sum_{p\leq x}f(p)$ is intimately related to the analytic properties of the Dirichlet series $\sum_{p\leq x}f(p)p^{-s}$. If $f(p)$ equals the von Mangoldt function, perhaps twisted by some automorphic data like values of a Dirichlet character or Hecke eigenvalues of a cusp form, then $\sum_{p\leq x}f(p)p^{-s}$ equals, up to a Dirichlet series supported on higher prime powers, the negative logarithmic derivative of an automorphic $L$-function $L(s)$, as follows from the Euler product decomposition of $L(s)$. The function $L(s)$ itself is nice, in addition to being an Euler product: it has a meromorphic continuation with at most a few poles and good behavior in vertical strips, has a functional equation, and conjecturally all its zeros lie on the axis of symmetry (Generalized Riemann Hypothesis). The logarithmic derivative $L'(s)/L(s)$ is therefore also meromorphic with simple poles at the zeros and poles of $L(s)$.

In the case of $f(p)=\Lambda(p)$, we have $L(s)=\zeta(s)$ so that there is a simple pole at $s=1$ with residue $1$ and also at every zero of $\zeta(s)$ with residue equal to the negative multiplicity of the zero. Conjecturally the zeros all lie on $\Re(s)=1/2$ which makes $\sum_{n\leq x}\Lambda(n)$ change rather smoothly: it is $x+O(x^{1/2+\epsilon})$. Of course $\sum_{p\leq x}\log(p)$ differs from $\sum_{n\leq x}\Lambda(n)$ only by $O(x^{1/2+\epsilon})$.

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I would say that the von Mangoldt function $\Lambda$ is more interesting than other multiplicative functions because of the relation $log = 1\star \Lambda$, where $\star$ is the convolution of arithmetic functions. This relation encodes the fundamental theorem of arithmetic, so "heuristically" it is of great interest, compared to the other propositions of the OP. – Bernikov Jul 21 '11 at 2:23
@Bernikov: The von Mangoldt function is not multiplicative (it is supported on prime powers). – GH from MO Jul 21 '11 at 9:59

It seems to me that the second Chebyshev function $\psi(x) = \sum_{n \le x} \Lambda(n)$ (where $\Lambda$ is the von Mangoldt function) is more natural, but as Gjergji says the two approximate each other. This is because the behavior of the second Chebyshev function is related by certain general theorems that I'm not familiar with to the behavior of the Dirichlet series

$$\sum_{n \ge 1} \frac{\Lambda(n)}{n^s}$$

and this Dirichlet series is precisely $\frac{-\zeta'(s)}{\zeta(s)}$, or the negative logarithmic derivative of $\zeta(s)$. This has the following intuitive interpretation: if we think of $\zeta(s)$ as the partition function

$$\zeta(s) = \sum_{n \ge 1} e^{-s \log n}$$

of the Riemann gas, then the negative logarithmic derivative of a partition function describes the expected value of energy at a given temperature, a fundamental property.

Edit: This observation goes at least as far back as Mackey, Unitary Group Representations in Physics, Probability, and Number Theory (1978), who wrote:

...Our main point here is that one could have been led to the main outline of the proof of the prime number theorem by using the physical interpretation of Laplace transforms provided by statistical mechanics. In particular, the function $-\frac{\zeta'}{\zeta}$ whose representation as a Dirichlet series (Laplace transform with discrete measure) plays a central role in the proof has a direct physical interpretation as the internal energy function.

Regarding why it is natural to assign the state $n$ the energy $\log n$, the point is that for $s > 1$ we get the only probability distributions on the natural numbers which satisfy certain natural properties; see this blog post, for example.

Of course a basic purely mathematical reason to consider the logarithmic derivative is the ideas around the argument principle.

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Partition functions? Riemann gases? Since when do these give "intuitive interpretations" of the zeta function? :) – David Hansen Jul 19 '11 at 4:52
@David: well, I suppose it is a matter of whether you care about physical intuition or not. The point is that the statistical properties of the Riemann gas should be closely related to statistical properties of the primes. It should be possible to write a complete proof of the PNT in which every step is physically motivated like this but I don't know if anyone's actually done it... – Qiaochu Yuan Jul 19 '11 at 5:11
I truly don't find an explanation involving "Riemann gasses" to be "physically motivated". Forcing a translation into the language of physics is not a motivation. – Greg Martin Jul 19 '11 at 9:13
@Greg: "forced" seems a little harsh to me. Some people take ideas like these quite seriously, and this exact observation was pointed out by no less a mathematician than Mackey: – Qiaochu Yuan Jul 19 '11 at 12:36
@Qiaochu: That's a good point! I meant nothing rude by my remarks - I'd simply like to see some evidence that this is a fruitful point of view, and for me "evidence" means "proofs of desirable theorems". – David Hansen Jul 19 '11 at 19:35

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