32 problems

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…

Local Bloch–Kato conjecture. The element

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 $f$ be a modification of a primitive Dirichlet character $\chi$ with finite modification set $S$: thus $f$ is completely multiplicative, $f(p)=\chi(p)$ for primes $p\notin S$,…

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 $k\geq 2$ be an integer, let $\chi$ be a real Dirichlet character, let $g_\chi:\mathbb{N}\to\{-1,1\}$ be the associated modified completely multiplicative character, and let…

Character-sum cancellation conjecture. For every $\varepsilon>0$ and every $x>q^{\varepsilon}$,

Let $p$ be an odd prime and let $\chi$ be a non-trivial even character of the first kind, written as $\chi=\theta\omega^i$, where $p$ does not divide the conductor of $\theta$ and…

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 $\mathcal F_3(X)$ be the family of primitive cubic characters over $\mathbb Q$ with conductor bounded by $X$. The constants $C_{\max}$ and $C_{\min}$ are those specified in the…

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…

Klurman–Tenenbaum–Teräväinen conjecture. The partial sums satisfy

Let $q$ and $x$ tend to infinity, and let $\chi$ range over Dirichlet characters modulo $q$. Let $\mu$ be the Möbius function, let $\lambda$ be the Liouville function defined by ……

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…

Prime number race mod $4$ conjecture.

Let $E$ be the elliptic curve under consideration, and for each $d$ let $\mathcal{C}_d$ be the set of even primitive Dirichlet characters of order $d$. Define … which the source id…

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…

Theta-function non-vanishing conjecture. For every non-trivial character $\chi$ modulo the prime $p$,

Let $f:\mathbb{N}\to\mathbb{C}$ be completely multiplicative. Suppose that $f$ takes only finitely many values, vanishes at only finitely many primes, and has bounded discrepancy,…

Generalized Ramanujan polynomial conjecture. The polynomial $R_k(z;\chi,\psi)$ is unimodular: all its roots lie on the unit circle.

Density conjecture. As $m$ tends to infinity, the family average converges to the unitary density:

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

Let $\chi$ be a nontrivial primitive Dirichlet character, and let $L(s,\chi)$ be its Dirichlet $L$-function, initially defined for $s=\sigma+it$ with $\sigma>1$ by … The critical l…