<http://arxiv.org/abs/0908.4287>
I could not find any reviews for it, but if true its a major claim, because it says that $\Re(\rho) < 21/40$ where $\rho$ is a zeta zero.


My question:



> 
> Are there any reviews of this paper, that reject or accept the claims made in this paper?
>  Any references will be highly appreciated.
> 
> 
> 



A note by N. A. Carella on zero-free regions
Well,


In this case it is a short elementary paper (6 pages) and it is easy to verify, although just the length of the paper and the importance of the result and some other aspects of the paper, such as the fact that it is in general mathematics part of arxiv and it does not seem as the autor uses latex is sufficient to be quite sceptical about the claim. It is easy to find mistakes. The paper has 6 pages. The first part seems to contain some standard results in the area. The proof of the new claimed very strong result, Theorem 1 is just one page long, and it starts on page 4, so it is sufficient to read that part. The author tries to use the fact that
$$
 \theta(x+y)-\theta(x)>0,
$$
where
$$ \theta(x)=\sum\_{ p < x } \log(p),$$
whenever $y \gg x^{21/40} $, which is a nice result of Baker-Harman-Pintz to prove the claimed zero free region. This is theorem 1 in the paper. It is easy to see that the claimed proof gives no such result. The author gives the elementary inequality
$$ \theta(x+y)-\theta(x) < y (\log x)^3.$$ This is certainly true in relevant ranges of $x$ and $y$. However the author then somehow uses this inequality in the wrong direction into something that essentially boils down to (Eq 4 in the paper does not hold in general)
$$ 0< \theta(x+y)-\theta(x) < y (\log x)^3 < \theta(x+y)-\theta(x). $$
Since this is obviously false the author obtains several contradictions, one of which is supposed to prove Theorem 1 (at least that is my guess how the author comes to that conclusion. The "proof" is not clear at all).


