10 problems

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 $f^*(n)=f(n)\mathbf 1_{\operatorname{gcd}(n,r)=1}$…

Ratios Conjecture. Under these constraints,

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 $\eta>0$,

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 $q$ 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…