# About Goldbach's conjecture : a certain differential equation

This question is related to https://mathoverflow.net/questions/61842/about-goldbachs-conjecture

Let $y(x)$ be a function such that $\alpha_{n}=O(y(n))$ . taking $y(x) : =x^{1/2}\log^{2} x$ leads to $y'=\dfrac{1}{2}x^{-1/2}\log^{2}x+2x^{-1/2}\log x$ . As $1/ \log x$ is roughly equal to $\pi(x)/x$ , is there any reason to believe that $\alpha_{n}=O(z(n))$ where the function $z(x)$ fulfills the differential equation $z'=\dfrac{z}{2\Delta}+z\dfrac{\pi(x+\Delta)-\pi(x-\Delta)}{\Delta}$ where $\Delta : =\Delta(x)$ is such that $\Delta(x)\leq (1+o(1))x$?

• why ? ${}{}{}{}{}$ – reuns Mar 15 '17 at 12:11
• Because. In case you wouldn't have noticed, I think induction can be used in mathematics to lead to discovery. – Sylvain Julien Mar 15 '17 at 12:15
• Goldbach is extremely complicated to analyze. You didn't define $\alpha_n$. Your differential equation is a mess : do you really think we can do anything better than replacing $\pi(x+a)-\pi(x)$ by $\frac{a}{\ln x} + O(a^\theta)$ and "hope" it will give something acceptable ? – reuns Mar 15 '17 at 12:57
• $\alpha_{n}$ is an error term defined in the link I gave. – Sylvain Julien Mar 15 '17 at 13:06
• In a more rigorous setting, one can consider the sequence of differential equations $y'_{k}=\dfrac{y_{k}}{2x^{1-1/k}}+y_{k}\dfrac{\pi(x+x^{1-1/k})-\pi(x-x^{1-1/k})}{x^{1-1/k}}$ and then take the limit as $k$ tends to $\infty$ . – Sylvain Julien Mar 15 '17 at 13:44