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Inspired by the previous answer I am thinking about simplest possible ways to prove $$ gcd\left(\binom n1_q,...,\binom n{n-1}_q\right)=\Phi_n(q). $$ One most primitive way to do it is to check that bo …

I got to the 225th prime with quick-and-dirty Mathematica code beta[n_]:=With[{p=Prime[n]}, With[{l=PrimitiveRootList[p]}, Module[{i,xl={},g}, For[i=1,i<=Length[l],i++, g=l[[i]]; Module …

Just an illustration - for $m$ up to 6000, $r$ up to 120. Quite mysterious looking, I would say. Version 2: in the coordinate system suggested by the answer of Lucia it looks somehow more regular, …

Certainly $$ \psi(1+z)=−\gamma+\sum_{k=2}^\infty(−1)^k\zeta(k)z^{k−1} $$ and $$ \psi(z)=−\gamma−\frac1z+\sum_{k=1}^\infty\frac z{k(k+z)} $$ are classical - see e. g. DLMF As for Mathematica, it answe …

For $n\geqslant0$ let $F_n(t)=\sum_{m\in\mathbb Z}a_{n,m}t^m$, where we are going to define $a_{n,m}$ for negative $m$ in such a way that $a_{n+1,m}=\frac{a_{n,m-1}+a_{n,m+1}}2$ for all $n\geqslant0$ …

The question is motivated by yet another possible approach to a combinatorial problem formulated previously in "Special" meanders. I'm not giving details of the connection as I believe the question is …

NOT AN ANSWER, just an illustration :D ($g$ up to $n=150$) Quite amusing... In case anybody wants to play with this, here is the Mathematica code g[n_] := Dot @@ Transpose[FactorInteger[n]] Graph …

Just out of curiosity - here is the plot of $\frac{p_{n+h}-p_n}h$ for $1\leqslant n,h\leqslant500$

For arbitrarily small $\varepsilon$, in a small enough neighborhood of any rational $\frac pq$ (more precisely, for $z=\varepsilon+i(\frac pq\pm\varepsilon')$) one starts seeing regular behavior, so w …

Actually I came upon this through MO a couple of days ago: in here (http://aimath.org/aimnews/lmfdb/) there is a mesmerizing image The caption reads A Maass form, one of the 20 different types of ob …

At least some part of the features may be explained by plotting $\frac1{r(n)}$, it looks like this: It is more or less clear that the slopes are $\frac1{k^2}$, $k=1,2,3,...$ (So the original plot …

Contemplating a question on math.SE, I have stumbled on this: Here, the point labeled $n$ is that root of the $n$th Bernoulli polynomial which has smallest positive imaginary part. Does anyone …

Visually, where black dots are values of $\zeta\left(\frac12+it\right)$ and the red ones are at the distance $\frac14+t^2$ from the origin.

Not an answer - only posting here since no pictures can be placed in comments That the $n$th row of the above picture has black pixel at the $m$th position means that the $m$th prime has odd multipli …

This is inspired by the Question on coefficient of $\exp(H_n).\log(H_n)$ in Lagarias equivalence of RH , an answer and some comments there. For $n\geqslant2$ denote $$ L(n):=\frac{\sigma(n)-H_n}{\exp( …