10 problems

Let … be the unique normalized cusp form of weight $12$ for $SL(2,\mathbb{Z})$, with Fourier coefficients $\tau(n)$. Lehmer's conjecture. … The conjecture asserts that the Ramanuja…

Let $\tau(n)$ denote the $n$th Fourier coefficient of Ramanujan's discriminant modular form. A prime occurs as a tau-value up to sign if $|\tau(n)|=\ell$ for some integer $n\geq 1$…

Zagier's conjecture. The function $c_0(y)$ has an oscillatory behaviour as $y\to0^+$.

Value-surjectivity question. For every integer $\alpha$, does there exist an $n\in\mathbb{Z}^+$ such that $\tau(n)=\alpha$?

Prime-value conjecture. It is conjectured that $|\tau(n)|$ takes on infinitely many prime values, and that the smallest such value corresponds to

Let $k\geq 1$ and let $a_1,\ldots,a_k$ be distinct non-negative integers such that $\tau(a_j)\ne 0$ for every $j$. Serre's conjecture. The set … is infinite. This is presented as a…

Let $p^n\geq 2$ be any prime power, and let $\varepsilon>0$ be arbitrarily small. The Ramanujan tau function $\tau$ is defined by the coefficients of the discriminant modular form…

The abc-conjecture. For every $\varepsilon>0$, there exists a constant $C_{\varepsilon}$ such that for all nonzero coprime integers $a,b,c$ satisfying $a+b=c$,

Let the Ramanujan $\tau$-function be defined by … and extend it to nonzero integers by $\tau(n):=\tau(|n|)$ and set $\tau(0):=0$. For $k\geq 0$, write $\tau^k$ for the $k$-fold ite…

Let $Q$ denote the first quadrant of the complex plane, and let $\Xi_{\Delta,N}(z)$ be the truncation after $N$ terms of the hyperbolic-gamma-function series defining the Ramanujan…