167 problems

Conrey–Farmer–Keating–Rubinstein–Snaith conjecture. The asymptotic formula

Let $\beta>0$, $k\in\mathbb{N}$, and $s\in\mathbb{R}_+$. Let $\mathsf{M}^{(\beta)}_N(k;s)$ denote the corresponding C$\beta$E field moment, and let $\mathfrak{c}^{(\beta)}(k;s)$ be…

Let $g_k$ denote the universal factor in the conjectural leading constant for the $2k$th moment of the Riemann zeta function. Conrey–Gonek conjecture. The eighth moment constant is…

High-moment asymptotic conjecture. Under these conditions,

Let $I_k(T)$ denote the $2k$th moment of the Riemann zeta function, and write its conjectural leading constant as $C_k=g_k a_k/\Gamma(k^2+1)$, where $a_k$ is the arithmetic factor.…

Let $n_1,n_2$ be non-negative integers, let $c(n_1,n_2)$ denote the leading-order coefficient in the mixed second-moment asymptotic for the relevant family of quadratic Dirichlet…

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

Murty–Murty moment conjecture. For every integer $k\ge2$, there exists a constant $C_k>0$ such that

Folklore conjecture. As $T\to\infty$, there is a positive constant $C_k$ such that

Let $P_X(s)$ and $Z_X(s)$ denote the prime and zero factors in the hybrid model for the Riemann zeta function, so that their product models $\zeta(s)$. Gonek–Hughes–Keating splitti…

Let $k$ be a positive integer, let $\zeta^{(k)}$ denote the $k$-th derivative of the Riemann zeta function, and let $a_k$ be the arithmetic factor appearing in the conjectural form…

Shifted-prime moment conjecture. There exists a constant $c_k(a,b)>0$ such that

Effective non-vanishing conjecture. One has

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…

Catalan-number odd-moment conjecture. The average value of the main term of the $(2k+1)$st moment is

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

Consider the family of quadratic Dirichlet $L$-functions $L(s,\chi_d)$, where $\chi_d$ is a quadratic Dirichlet character, and let $g_k$ denote the constant in the corresponding co…

Conrey–Ghosh conjecture. The parameter $\theta=1$ is permissible in both auxiliary mean formulas (20) and (21).

Moment conjecture for the zeta function. There is a constant $c_k$ such that, as $T\to\infty$,

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…

Moment asymptotic conjecture. For $k=1,2$ there exists a constant $A_k>0$ such that

Let $Z_X(s)$ be the truncated Hadamard factor, let $X,T\to\infty$ with $X=O((\log T)^{2-\epsilon})$, and let $k>-1/2$ be fixed. Let $G$ be the Barnes $G$-function and let $\gamma$…

Let $g(t)=f(t/T)$, where $f:\mathbb{R}^+\to\mathbb{R}$ is non-negative, bounded, and integrable. Let $P_k$ be the degree-$k^2$ polynomial defined by the stated $2k$-fold residue, w…

Let … where $P_4$ is a polynomial of degree four and $E_2(T)$ is the error term. Conjecture on the fourth-moment error term. For every $\varepsilon>0$, … The source further states…