Has anyone noticed the paper On the zeros of the zeta function and eigenvalue problems by M. R. Pistorius, available on ArXiv? The author claims a proof of RH, and also a growth condition on the zeros. It was posted two weeks ago, and I expected it would have been shot down by now. Has there bee...

I would like to show that$$\sum_{j=n-k}^n\binom nj(1-x)^{n-j-1}x^{j-1}(j-nx)=\binom n{n-k}(n-k)(1-x)^kx^{n-k-1}\\\sum_{k=0}^{n-1}\frac{(-1)^k}n\binom{n-1}k\binom n{n-k}(n-k)(1-x)^kx^{n-k-1}\\=(-1)^{n-1}\sum_{k=0}^{n-1}\binom{n-1}k\binom{n+k-1}k(-x)^k$$ I feel I exhausted all identities/properties...

What simplification I can do for this problem? I tried to use $\sin a\sin b=-2(\cos (a+b)+\cos (a-b))$, but I didn't got it...

First we define $f:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{R}$ by$$f(x,a)=\begin{cases}\dfrac1a;&x=1\\\displaystyle\sum_{k=1}^{a-x+1}\frac1a f(x-1,a-k);&1<x\leqslant a\end{cases}.$$ ($f$ is defined that way only where $x\leq a$ in every other place it is $0$.) How can I find $$\sum_{k=1}^ak...

Do there exist a positive integer $N$ and three primes $p_1<p_2<p_3$ such that the only prime factors of $N-p_1$ is $p_2$, the only prime factors of $N-p_2$ are $p_1,p_3$ and the only prime factors of $N-p_3$ are $p_1,p_2$?

I know that the general expression for polynomial with degree $n$ is $a_n x^n + a_{n-1}x^{n-1} + \cdots + a_0$. My question is, is there any specific name to linear combination of $x^{\text{rational number}}$?. For example $2 x^{2.5} + 1.3 x^{1.3}$. Notice that we have a similar question here, bu...

$\sum \limits_{n=1}^{\infty}$ and $\sum \limits_{n>0}$ here $n\in \mathbb{N}$ Sometimes i stuck with the notations of summation. are these two equivalent

Let $p$ be a prime number, $P$ and $Q$ monic polynomial of degree $k\geq 2$, irreducible over $\mathbb Z_p$. Do we have $$\mathbb Z_{p^2}[x]/(P) \simeq \mathbb Z_{p^2}[x]/(Q)\ ?$$