14 problems

Let $k\geq 2$ be an integer, let $\chi$ be a real non-principal Dirichlet character modulo $q$, and let $f=\mu^{(k)}\chi$ or $f=\mu^{(k)}g_\chi$. Then the large-values limsup conje…

Conjecture $C_n$. For every $\varepsilon>0$, there exists $\delta=\delta(n)>0$ such that, for all $M$,

Let $\chi_1$ and $\chi_2$ be nontrivial, primitive Dirichlet characters modulo $q_1$ and $q_2$, respectively, with $\chi_1\chi_2(-1)=1$. Let $F_{\chi_1,\chi_2}$ be the smallest num…

Let $\chi_1$ and $\chi_2$ be nontrivial, primitive Dirichlet characters modulo $q_1$ and $q_2$, respectively, with $\chi_1\chi_2(-1)=1$. Let $S_{\chi_1,\chi_2}$ be their generalize…

Let $n$ be a positive squarefree integer, let $d>1$ be an integer not dividing $n$, set $d_1=\gcd(d,n)$, and let … Let $F_n$ be the Fekete polynomial and let…

Let $\Psi_6(X)$ be the primitive Dirichlet characters of order $6$ with conductor at most $X$, and let $\tau(\chi)$ be the Gauss sum of $\chi$. Sextic Gauss-sum equidistribution co…

For odd integers $k$, consider primitive quartic Dirichlet characters $\chi$ of odd conductor $q$, and let $\tau(\chi)$ denote their Gauss sums. Odd-moment equidistribution conject…

Klurman–Tenenbaum–Teräväinen conjecture. The partial sums satisfy

Let $\eta$ be a real number satisfying $0<\eta\leq 1$. A real character is a Dirichlet character taking real values, and a character is non-principal when it is not the principal c…

Let $E$ be the elliptic curve under consideration, let $\mathcal{C}_p(X)$ denote the even primitive Dirichlet characters of order $p$ and conductor at most $X$, and define…

Square-root cancellation conjecture. As $N\to\infty$,

Let $\chi$ be a Dirichlet character, and let $n_\chi$ be the first natural number such that $\chi(n_\chi)\neq1$. Character least nontrivial value conjecture. For every fixed…

Let $\xi$ be an odd primitive Dirichlet character modulo $k$ with $3\nmid k$, and define … Let $n_\xi$ denote the least positive integer $n$ with $\xi(n)\neq 1$, and let $M(\chi)$…

Let $\varphi\in S_k(N,\omega)$ be a primitive form, let $\eta$ be a primitive Dirichlet character, and let $L_f(s,\operatorname{Sym}^n\varphi,\eta)$ denote the finite $L$-function…