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Some general information on the problem, which is probably open. As pointed out by Ivan Neretin at math.stackexchange, this is OEIS A100129. There you can find the first 16 numbers with this property …

The basic idea is that sometimes arithmetic objects aren't "nice enough" for the tools that support the Langlands program (representation theory, noncommutative harmonic analysis, etc) to work. They …

I think you can find the proof you want in: Andrew Ogg, On the Eigenvalues of Hecke Operators (1969) Perhaps you can find it more thoroughly explained in Shimura's book, or on more concrete notes …

The best lower bound known seems to be due to Ram Murty: $$\tau(n)=\Omega (n^{11/2}e^{c \log n / \log \log n})$$ for some $c>0$ absolute and effective (note: thanks to the Sato-Tate conjecture might …

This follows from the well-known Touchard's congruence (here for a random reference). Following your notation, the congruence is: $$B(n+p^k)\equiv kB(n)+B(n+1) \mod \ p$$ Taking $n=0$: $$B(p^k)\equ …

Actually Ramaré's paper is freely avaible at his Lille university page (link here). The estimate that he uses to deduce his result is $$\bigg|\sum_{n\leq x} \frac{\mu(n)}{n}\bigg|\leq \bigg(\frac{3} …

As far as I know, the precise conjecture for what you are asking is: All the elements of the Selberg class that are not Artin L-functions are: motivic L-functions of dimension bigger than 0. transc …

It is an open problem, only known for $k=1,2$. Both the conjecture and the known cases are due to Schinzel. You can find a nice survey here: "On the third iterates of the φ- and σ-functions" H. Maie …

Norman Do calculates somes $V_{g,n}(L)$ polynomials for small $g$ and $n$ in the appendix A of his thesis: Norman Nam Van Do, Intersection theory on moduli spaces of curves via hyperbolic geometry ( …

The answer seems to be no. It is hard to find direct confirmation, but every time their names are mentioned together it is made clear that there's no relationship. The Russian mathematician Askold …

The problem is that your $L_N(k,\xi\chi)$ do not lie in an extension of $\mathbb{Q}_p$. You can interpolate the algebraic part, but using the functional equation of the original Dirichelt L-functions …

All the sporadic groups except for $M_{23}$ and $M_{24}$ are realized over $\mathbb{Q}$ using the rigidity criterion. This technique is explained in all three main textbooks on inverse Galois theory: …

As mentioned in the comments this is essentially the classic problem of finding integer points of the Mordell curve, and a lot of work has gone into it (for example towards bounding the number of solu …

Given the importance of the proceedings, I leave a link to the copy at Library Genesis: A. Borel and W. Casselman (Editors) Automorphic Forms, Representations and L-functions (1979)

As mentioned in the comments, this is precisely Hilbert's twelfth problem, for the simple reason that any solution to that problem can be turned into a "group law argument" (or any solution to it is a …