In [Mean values of multiplicative functions over function fields](https://arxiv.org/abs/1504.05409) the mention a proof of Halasz inequality in one of their future pre-prints. In fact here is an a proof from [1999](https://arxiv.org/abs/math/9911246).


I would like to see a proof in any way shape or form, as it might lead to a proof of the prime number theorem.




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There are various statements floating around here are two:


[[1](https://terrytao.wordpress.com/2015/11/23/a-cheap-version-of-halaszs-inequality/),[3](https://terrytao.wordpress.com/2015/02/24/254a-supplement-6-a-cheap-version-of-the-theorems-of-halasz-and-matomaki-radziwill/)] Let $f : \mathbb{N} \to \mathbb{C}$ be multiplicative function such that $|f(n)| \leq 1$. Then if we establish for all real numbers $|t| < T$ that:
$$ \sum\_{p < x} \frac{1 - \mathrm{Re}[\, f(p) p^{-it} \,]}{p} \geq M$$
then the average value of $f(n)$ is quite small:
$$ \frac{1}{x} \sum\_{n < x} f(n) \ll (1+M)e^{-M} + \frac{1}{\sqrt{T}} $$
even the "cheap" version stated in the blog might already imply the prime number theorem.


[[2](http://www.dms.umontreal.ca/%7Eandrew/PDF/EdinLecture.pdf)] If the mean value of $f$ is "large" in absolute value, then $f(n)$ pretends to be $n^{it}$ for some "small" real $t$. Here, pretending is defined in terms of [Kullback-Liebler](https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) distance:


$$ \mathbb{D}(f, g; x)^2 = \sum\_{p \leq x} \frac{1 - \mathrm{Re}\big[\,f(p)\overline{g(p)}\,\big]}{p} $$


All of the proofs seem long and technical, and the new proof may not be much simpler. I still might like to know if the weaker version implies PNT



new proof of Halasz inequality
Heuristically (and in fact rigorously with a bit of work) we have
$$
\sum\_{p \leq x} \frac{1 + \Re(p^{it})}{p} = \log\log x + \Re \log \zeta(1 + \frac{1}{\log x} + it) + O(1)
$$
Therefore if $\zeta$ has a zero at $1 + iu$ then the hyp othesis of Halasz's theorem fails and it doesn't imply the Prime number theorem. So you get the Prime Number Theorem from Halasz + non-vanishing of $\zeta(s)$ on the $1$-line. 


