123 problems

Let $\chi$ be a primitive quadratic Dirichlet character, and let $L(s,\chi)$ denote its Dirichlet $L$-function. Chowla's conjecture. … The conjecture asserts non-vanishing at the c…

Let $\chi\pmod{q}$ be a Dirichlet character, and write $s=\sigma+it$. Generalized Lindelöf hypothesis. For every $\varepsilon>0$, if $\sigma\geq 1/2$ and $|s-1|\geq 1/2$, then … Th…

CFKRS eighth-moment conjecture. As $q\to\infty$,

Generalized Riemann Hypothesis. All zeros of Dirichlet $L$-functions lie on the critical line

Nonvanishing conjecture. For every character $\chi$ and every real $\sigma$ with

Let $q$ be a modulus, let $\boldsymbol\alpha=(\alpha_1,\alpha_2,\alpha_3)$ and $\boldsymbol\beta=(\beta_1,\beta_2,\beta_3)$ be sufficiently small shifts, and let…

Andrade–Jung–Shamesaldeen moment conjecture. For any $k\in\mathbb{N}$,

Let $d\in\mathcal{D}$ be a positive discriminant, let $\chi_d=\left(\frac{d}{\cdot}\right)$ be the associated Kronecker character, and let $L(s,\chi_d)$ be its Dirichlet $L$-functi…

For every fundamental discriminant $D\in\mathbb Z$, let $\Lambda_D$ be the associated De Bruijn–Newman constant. Weak Newman conjecture. … where the supremum is taken over all fund…

Let $q$ be a modulus, let $f$ be the test function used in the 1-level density, and let $D_{1;q}(\widehat{f})$ denote the corresponding density with scaling parameter $Q=q$. Write…

Let $\chi$ be a primitive character modulo $q$, and define its Dirichlet $L$-function by … The Extended Riemann Hypothesis. The function $L(s,\chi)$ has no zeros with real part dif…

Let $q$ be an integer with $q \not\equiv 2 \pmod{4}$, and let $\chi$ range over primitive even characters modulo $q$. Define … where $\phi^*(q)$ is the number of primitive characte…

Let $p$ be prime, and let the sum run over Dirichlet characters modulo $p$. Ramachandra's conjecture. … This predicts the leading asymptotic for the fourth moment in the family of…

Hybrid product conjecture. Let $k\geq0$ be any real number. Suppose that $X$ and $q\to\infty$ with $X\ll(\log q)^{2-\epsilon}$. Then

Random-matrix conjecture for $Z_X$. Let $k>-\frac{1}{2}$ be any real number. Suppose that $X,q\to\infty$ with $X\ll(\log q)^{2-\epsilon}$ and $w(q)\ll X^{1+\epsilon}$, where $w(q)$…

The Dirichlet moment conjecture. For $k$ fixed with $\operatorname{Re}k\geq0$,

Let $\chi_d$ range over the real quadratic Dirichlet characters appearing in the family considered in the paper. Quadratic-character maximum-value conjecture. … Random matrix theor…

Gramain's conjecture.

Extreme-value conjecture. Given the extraordinary decay of $\Phi(\tau)$ in the relevant range, the above estimate should hold, at worst with a slightly different constant.

Let $\chi_d$ be a real primitive Dirichlet character modulo $d$, let $D^*=\sum_{|d|\leq D}1$, and let $a_4$ be the arithmetic factor in the conjectural moment formula. Soundararaja…

Let $\chi$ be a primitive Dirichlet character of modulus $r$, let $L(s,\chi)$ be its Dirichlet $L$-function, and let $\xi(s,\chi)$ be the associated completed function. For…

Let $g(u)$ be a suitable weight function, and let $\mathcal F$ be the family of real Dirichlet $L$-functions $L(s,\chi_d)$ associated with negative fundamental discriminants $d<0$.…

Universal dominance conjecture. For any modulus $N\ge 3$, the quantity $\log L(1,\chi_{1,a})$ achieves its unique absolute minimum over all non-principal residue classes…

Let $q\not\equiv2\pmod4$, and let $\varphi^*(q)$ denote the number of primitive characters modulo $q$. For each primitive character $\chi$ modulo $q$, write the non-trivial zeros o…

Let $\chi$ be a primitive Dirichlet character modulo $q>1$, and let $L(s,\chi)$ be its Dirichlet $L$-function, with non-trivial zeros $\rho=\tfrac12+i\gamma$. Assume the Generalise…