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By Cesaro-Stolz theorem, we have that $\displaystyle\lim_{n\to\infty}\frac{1}{n^2 \log(n)}\sum_{k=1}^{n} k\log(k)=\frac{1}{2}$ and then we see
that for $n$ large, we have that $\displaystyle\frac{1}{n^2}\sum_{k=1}^{n} k\log(k)\approx\frac{\log(n)}{2}\Rightarrow \frac{-4}{n^2}\sum_{k=1}^{n} k\log(...
I was just wondering if the following is correct to tackle the Goldbach Conjecture?
Let $G(x)$ be the Goldbach function and $p\in\mathbb{P}$ such that
$$\int_{2}^{x}\left[x\left(\cfrac{p+p}{p}\right)\right]dx=G(x)$$
Using the integral definition of the arctangent function, we may write $$\arctan{\left(\sin{x}\right)}=\int_{0}^{1}\mathrm{d}y\,\frac{\sin{x}}{1+y^2\sin^2{x}},$$
thus, transforming the integral into a double integral. Changing the order of integration, we find:
$$\begin{align}
\mathcal{I}
&=\in...
