15 problems

Let $K/\mathbb{Q}$ be a finite Galois extension, and let $\zeta_K(s)$ be its Dedekind zeta function. Its non-trivial zeros are the zeros other than the trivial zeros arising from t…

Let ${\mathbf L}/{\mathbf K}$ be an extension of number fields, and let $\zeta_{\mathbf K}(s)$ and $\zeta_{\mathbf L}(s)$ denote their Dedekind zeta functions. Dedekind's conjectur…

Let $\mathrm{K}$ be an algebraic number field of degree $n>1$ over $\mathbb{Q}$, let $\mathfrak{o}=\mathfrak{o}_{\mathrm{K}}$ be its ring of integers, and let…

Langlands reciprocity conjecture. For any fixed real $k>0$,

Let $K$ be a number field, and let $\zeta_K(s)$ denote its Dedekind zeta function. The zeros of $\zeta_K(s)$ in the critical strip are the zeros with $0<\operatorname{Re}(s)<1$. Si…

For imaginary quadratic fields, let $P_M(-d)$ count primes less than $M$ that split completely and whose factors lie in the cone $C_1$. Let…

Let $K$ be a number field of degree $n$, and let $eta_K(s)$ denote its Dedekind zeta function. Its nontrivial zeros are the zeros in the critical strip $0<\Re s<1$. Generalized Rie…

Let $\chi$ be a primitive Dirichlet character, and let $K$ be an Abelian number field. Write $\rho=\beta+i\gamma$ for nontrivial zeros of the relevant Dirichlet $L$-function…

Extreme-residue conjecture. The true upper and lower bounds are, respectively,

For a number field $E$ of degree $n$, let $Q_E$ be the positive definite quadratic form on $\mathcal{O}_E/\mathbb Z$ given by … and let $m_1(E)$ be its first successive minimum. Th…

Let $E$ range over totally real cubic number fields, and let $m_1(E)$ denote the first successive minimum of the positive definite quadratic form $Q_E$ on $\mathcal{O}_E/\mathbb Z$…

Let $\mathbb{K}$ be a quadratic extension, let $\chi$ be the associated real Dirichlet character, and let $a(k)$ be the arithmetic factor defined by the Euler product in the paper.…

Let $\mathbb{K}$ be a Galois extension of $\mathbb{Q}$ of degree $n$, and let $Z_\mathbb{K}(s,X)$ be the truncated product over the non-trivial zeros of $\zeta_\mathbb{K}(s)$. Writ…

Let $K$ be a number field, let $\xi_K$ denote its Dedekind xi-function, and define $\kappa_n$ as the coefficients in the series expansion of … The formulas established formally for…

Algebraic conjecture for $\eta_k$. The trivial Riemann hypothesis implies the generalized Riemann hypothesis for $\eta_k$: