# An upper bound for the number of divisors?

I have a diploma in math, but have never published anything. Now I am doing math in my spare time and came to the following result. Now I want to know if there is interest in this result or not, so to publish or not? Let $\tau(n)$ be the number of divisors of $n$, $(n,l)$ be the $\gcd(n,l)$ and $H_n$ be the $n$-th harmonic number. Then I believe I can prove that: $$\tau(n) \le \frac{1}{n}\sum_{0\le l \le n-1} \bigg( H_{(n,l)} + \exp(H_{(n,l)})\cdot \log(H_{(n,l)}) \bigg)$$ is equivalent to Riemann hypothesis. The proof makes use of Lagarias inequality and of some group theoretic result. What do you think, is there interest in this kind of thing or not and how should I go on when I want to publish?

Edit: I wrote the notes down. It would be nice if someone interested takes the time to read it and gives me feedback. Unfortunately I can not upload to arxiv.

• This request could fall into the crank camp or the innocent camp. I will assume innocent camp for now. There could be interest in the result, enough that you could submit a preprint to ArXiv. If you do not do that, you could write email to Lagarias or those using his result. Asking here is sorta OK, but asking more pointedly who would be interested (so you could write to them ) is better. At the least, someone here might know of someone and can point that person to this question (and your writeup, when you have one). Gerhard "But Make It Less Solicitous" Paseman, 2018.05.08. – Gerhard Paseman May 8 '18 at 21:34
• ok, thanks. I will write him an email. – orgesleka May 8 '18 at 21:36
• Good. Respect his (and others) time, and ask for what you want. (Be prepared to hear no.) It is reasonable to ask not only if he is interested, but would like to read your write up. More importantly, if there is someone else who may be interested, can he forward your request to (or give you an address for) that person. If your request is clean, direct, honest, and briefly handled, it may still go unappreciated, but you now have an approach to try on the next contact. (I know of no one in my circle who shows such interests at present.) Gerhard "Linked Lists Can Be Long" Paseman, 2018.05.08 – Gerhard Paseman May 8 '18 at 21:53
• How sharp is this upper bound? That should give you an idea as to whether the result is publishable or not. – Jose Arnaldo Bebita-Dris May 9 '18 at 3:09
• @stackExchangeUser: It seems to me that your result (assuming it is correct) ir just a small variation on a theme by Lagarias and it doesn't bring any new insight into the understanding the Zeta function. I would say "not publishable". – Alex M. May 9 '18 at 15:59