Chebyshev's first function prime count

Question

How is Chebyshev's first function $$\vartheta(N)=\sum_{p\leq N}\log p$$ useful in counting primes? Can it alone be used to analytically derive the prime number theorem?

Answer 1

The function $\sum \log p$ is useful for counting primes because it only increases at primes. This property is used for example in Bertrand's theorem, the idea being roughly that if $\vartheta(2x)>\vartheta(x)$ then there must be a prime between $x$ and $2x.$

Proofs of the PNT tend to be variations on the analytic proofs of Hadamard/de la Vallee Poussin or the "elementary" proof of Selberg/Erdős. There may be other proofs but I doubt any proof can be said to rely solely on use of $\vartheta(x).$

Harold Edwards in Riemann's Zeta Function gives the proofs of both Hadamard and de la Vallee Poussin. G.J.O. Jameson gives Selberg/Erdős' "elementary" proof in Chapter 6 and two analytic proofs in Chapter 3 of The Prime Number Theorem. Ingham presents an analytic proof at p. 26 of The Distribution of Prime Numbers. Apostol in Introduction to Analytic Number Theory gives an analytic proof in Chapter 13. Davenport in Multiplicative Number Theory at page 111 also offers a version of the classic proof.

All of these proofs involve the log-prime counting function $\psi(x)$ but of course $\psi(x)$ can be expressed in terms of $\vartheta(x).$

Can $\vartheta(x)$ be used alone to derive the PNT? Hopefully the references here would allow you decide whether or not $\psi(x)$ (and implicitly $\vartheta(x)$) meet your criteria for being used alone.

Answer 2

You can see directly the relationship using partial summation. In fact, you have $$\underset{p\leq x}{\sum}\log\left(p\right)=\pi\left(x\right)\log\left(x\right)-\int_{2}^{x}\frac{\pi\left(t\right)}{t}dt$$ where $\pi\left(*\right)$ is the prime counting function.