I have an hypothesis and I tried a lot of prime numbers on it and it worked , so if this hypothesis is correct ; How can it contribute to math ?
- Let $y$ be a non-zero natural number;
- Let $p_y$ be the $y$th prime number, indexing from 1. E.g. $p_1 = 2$.
- Let $\sigma_1(y)$ be the divisor sum of $y$.
The hypothesis is
$$\frac{(p_y+y)p_y}{2 \ln\left(\frac{(p_y+y)p_y}2\right) \ln( k p_y^n )} \approx 1$$
where $k = 7.63575844860837$ and
$$n=\frac{(\sigma_1(y)+y)\sigma_1(y)+(p_y+y)p_y}{2\ln\left(\frac{(\sigma_1(y)+y)\sigma_1(y)+(p_y+y)p_y}2\right) \ln(kp_y)}$$
Per @Tony Mathew's comment, the hypothesis is equivalent to
$$\frac{\alpha_y}{\log{\alpha_y}} \approx \log{k} + \frac{\alpha_y+\beta_y}{\log{\left(\alpha_y+\beta_y\right)}} \frac{\log{p_y}}{\log{k p_y}}$$
where $$\alpha_y=\frac{\left(p_y+y\right)p_y}{2},\quad \beta_y=\frac{\left(\sigma_1(y)+y\right)\sigma_1(y)}{2}$$
Define $f(x)=\frac{x}{\log{x}}$. By the prime number theorem, $f(x)\approx \pi(x)$, where $\pi(x)$ is the prime counting function.
Now, the hypothesis is $$f\left(\alpha_y\right)\approx \log{k} + f\left(\alpha_y+\beta_y\right)\frac{\log{p_y}}{\log{kp_y}}$$
If we let $K=\log{k}$, this becomes $$f\left(\alpha_y\right)- f\left(\alpha_y+\beta_y\right)\frac{1}{1+\frac{K}{\log{p_y}}} \approx K$$
but for large enough $y$, $\frac{K}{p_y}$ will be negligible, leading to $$f\left(\alpha_y\right)- f\left(\alpha_y+\beta_y\right)\approx K$$
Now note that $f$ is increasing; so we would expect $K=\log{k}$ to be negative. The hypothesis now becomes a question about the number of primes between $\alpha_y$ and $\alpha_y+\beta_y$, but I don't know how to go any further with the particular forms these numbers take.