18 problems

Let … Assume that $\alpha,\beta,\gamma,\delta\ll 1/\log T$ and $\operatorname{Re}(\gamma),\operatorname{Re}(\delta)>0$. Farmer's ratios conjecture. As $T\to\infty$, … This predicts…

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 $q\equiv1\pmod 4$, let $\mathbb{P}_{2g+1}$ denote the monic irreducible polynomials $P$ of degree $2g+1$ over $\mathbb{F}_q[T]$, and let $\chi_P$ be the associated prime charac…

Let $T>0$, let $\rho(s)=\zeta(s)$ be the Riemann zeta function, and let $A_{\zeta}(\alpha,\beta,\gamma,\delta)$ denote the product over primes appearing in the ratios formula. Assu…

Fix a real $M\geqslant 0$. Let $\bm{a}\in \mathcal{O}^6$ and $d\in \mathcal{O}^+$ with $\lvert \bm{a}\rvert,\lvert d\rvert\leqslant \widehat M$. Let $\bm{b}\in K_\infty^6$ with…

Let $r$ be a prime, let $\chi_0$ be the principal character modulo $r$, and let $f$ be a completely multiplicative function with $f^*(n)=f(n)\mathbf 1_{\operatorname{gcd}(n,r)=1}$…

Let $K=\mathbb{Q}(i)$, let $\chi_{(1+i)^5\varpi}$ denote the quadratic Hecke character indexed by a prime element $\varpi$ satisfying $\varpi\equiv 1\pmod{(1+i)^3}$, and let…

Let $\mathcal{F}(K)=\{L_k(s):1\le k\le K\}$, let $M=\log K$, and let $R_K(\alpha,\gamma)$ be the ratios-conjecture prediction built from the Euler product $G(\alpha,\gamma)$. For s…

Let $K=X^\lambda$ with $0<\lambda<1$, and let $C_f$, $C_\Phi$, $C'_\Phi$, $\widetilde\Phi$, $C_{\Phi,\zeta}$, $C_{\Phi,L}$, and $A'_\Phi$ be the constants defined in the paper. Rat…

Let $f$ be an even Schwartz function, let $w$ be the weight function, and define the weighted ratio average … For $\operatorname{Re}(2\nu'+\nu)>-1$ and…

The CFZ–CS ratios conjecture. Suppose the sets $A,B,C,D$ are as above and the imaginary parts of all parameters are $O(T^{1-\xi})$ for some $\xi>0$. Then, for some $\eta>0$,

Let $\alpha,\beta,\gamma,\delta$ be complex parameters, let $T>0$, and let $\psi$ be holomorphic in a strip around the real axis and rapidly decreasing on the real axis. Define … w…

Let $F$ be a Selberg-class $L$-function, let $r_{F\otimes\overline F}$ be the residue of $(F\otimes\overline F)(s)$ at $s=1$, and let $A_F$ be the arithmetic factor defined in the…

Let $F$ be a Selberg-class $L$-function, let $R_F(\alpha,\beta,\gamma,\delta)$ denote the corresponding integral of ratios of logarithmic derivatives, and let…

Local ratios conjecture. For these fixed parameters,

Let ${\mathcal F}(X)$ be the second family of elliptic curves under consideration, with conductor parameter $C(t)$. Define … and let $A(\alpha,\gamma)$ be the analytic Euler produc…

Let ${\mathcal F}(X)$ be the family of elliptic curves under consideration, with analytic conductor $N_E$ and root number $\omega_E$. For complex shifts $\alpha$ and $\gamma$, defi…

Let $H_k^\ast(N)$ be the family of cuspidal newforms of weight $k$ and level $N$, let $R_{H_k^\ast(N)}(\alpha,\gamma)$ denote the corresponding ratios quantity, let $X_L$ be the fa…