Hi everyone


This is my first post...
I do mathematics from home... ie., not attached with any institution...
I have deduced some results...


$\lim \inf\_{n\to\infty} \frac{d\_n}{\log p\_n} = 0$


and, for constants $A,B$


$\lim\_{n\to\infty} \log p\_n - \sum\_{i=1}^{n-1} \frac{d\_i}{p\_{i+1}} = A$


$\lim\_{n\to\infty} \log p\_n - \sum\_{i=1}^{n-1} \frac{d\_i}{p\_{i}} = B$


Where, $p\_n$ is the nth prime... $d\_n = p\_{n+1} - p\_n$


My question is:
Do you think these results are good ?



A result on prime numbers
Disclaimer: I am no specialist in Analytic Number Theory, nor did I read the whole paper under the link. I just looked into the end of the argument, and there is a limit computation (10) there.


From what I know from Analysis, this computation is clearly wrong, not in the sense that the answer is necessarily wrong, but in the sense that the premises do not justify the conclusion. The author attempts to compute the lower limit of the product $$\liminf\_{n\to\infty}\left(\frac{p\_n}{\log p\_n}\log\frac{p\_{n+1}}{p\_n}\right)$$
as the product of the limits. He replaces the second factor with $log(1)=0$ and proceeds to claim that the lower limit of the product is $0$. However, even though the (lower) limit of the second factor may well be $0$, the limit of the first factor is clearly $\infty$, so one cannot compute the lower limit of the product in this way.


