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I am studying Zagier's paper titled Newman's Short Proof of the Prime Number Theorem. In the paper, proof ($\textbf{VI}$), we are trying to show that $\theta(x) \sim x$, where $\theta(x) = \sum_{p \leq x} ln(p)$. If I understand correctly, to do this, we take that fact that $\theta(x) = O(x)$ (as shown in proof ($\textbf{III}$)), meaning that $\lim_{x \to \infty} \theta(x) = c \cdot x$, and we then 'squeeze' $c$ from both directions, ultimately showing that $c = 1$.

My main confusion comes from line 3-4 of the proof, where the author shows that for $\lambda > 1$, and arbitrarily large $x$ so that $\theta(x) \geq \lambda x$, $$\int_x^{\lambda x} \frac{\theta(t) -t}{t^2}dt > 0, \ \ \ \ (1)$$ and then asserts that this is a contradiction for such $x$ by proof ($\textbf{V}$), which simply states that $\int_1^{\infty} \frac{\theta(x) - x}{x^2}$ is a convergent integral. I'm confused as to how (1) contradicts $(\textbf{V})$.

After poking around the internet, I believe that the contradiction not only has to do with ( V ), but it may also have to do with Newmans Tauberian Theorem, which the author refers to as the Analytic Theorem . With that being said, the author does not explicitly say he is using Newmans Tauberian Theorem, so I am not entirely sure.

Could anyone explain to me why (1) contradicts V ? Or, if not that, perhaps confirm or deny that we are using the Tauberian Theorem here? Anything to help point me in the right direction is greatly appreciated!

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