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Martin Sleziak
Martin Sleziak
07:10
I have noticed this new room on MO:

 Analytic number theory

For people interested in analytic number theory, a field of ma...
epic_math
epic_math
07:44
I found an interesting equivalent of the prime number theorem: $$\lim_{n\to\infty}\frac{\lambda(1)+\lambda(2)+...+\lambda(n)}{n}=0$$ where $\lambda$ is Liouville's function.
2
Martin Sleziak
Martin Sleziak
08:39
This seems to be one part of Theorem 4.4 in Apostol's book.
It says that PNT is equivalent to $$\lim_{x\to\infty} \frac{\psi(x)}x=1.$$
The function $\psi(x)$ is defined on page 75 ad $$\psi(x)=\sum_{n\le x}\Lambda(x).$$
Sorry, my bad - they are not the same. The one I mentioned has $\Lambda$ rather than $\lambda$.

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