60 problems

Goldston–Montgomery variance conjecture. For every fixed $0<\delta<1$, this variance satisfies

Let $\zeta(s)$ be the Riemann zeta function, and let $\gamma>0$ range over the positive imaginary ordinates of its nontrivial zeros $\rho=\beta+i\gamma$. Linear Independence Conjec…

Hardy Z-function joint moment conjecture. As $T\to\infty$,

Let … be the Riesz function. Riesz's criterion. The Riemann hypothesis is equivalent to the condition that, for every $\varepsilon>0$ and as $x\to\infty$ through real values, … Thi…

Let $\zeta(s)$ denote the Riemann zeta function. For fixed $V$, Selberg's central limit theorem describes the distribution of $\log|\zeta(1/2+it)|$ after normalization by…

Let $\mathbf{A}\in\mathbb{U}(N)$, and let $Z_{\mathbf{A}}(\theta)$ be the real-valued Hardy-type characteristic-polynomial function … For arbitrary positive real parameters $s$ and…

Assume the nontrivial zeros of the zeta function are written as $\rho_n=1/2+i\gamma_n$. LI conjecture. The imaginary parts $\gamma_n$ of the nontrivial zeros are linearly independe…

Let $\Delta(x)$ be the Dirichlet divisor error term, and define, for $t\geq2$, … where $\gamma$ is Euler's constant. Here $T$ is large and $U$ satisfies $1\ll U\ll T^{1/2}$. Jutila…

Turan's criterion. If every function $S_n(s)$, for $n=1,2,3,\ldots$, has no zeros in the half-plane $\sigma>1$, then the Riemann Hypothesis is true.

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

Let $c_k$ be the coefficients of the Riesz–Hardy–Littlewood wave, and define the critical function by … Here $\alpha$ and $\beta$ range over the allowed parameters, and the critica…

Let $c_k$ be the coefficients occurring in the Riesz–Hardy–Littlewood wave, and let $\zeta(s)$ denote the Riemann zeta function. The preceding discussion considers the limiting cas…

Quarter-power bounds conjecture. It is conjectured that

Pointwise bound conjecture for ${\Cal Z}_2$. The exponent $\frac{4}{3}(1-\sigma)$ should be replaceable by $\frac{1}{2}-\sigma$, so 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 $Z_X(s)$ be the truncated Hadamard factor in the hybrid product, formed from zeta zeros near height $T$. Replace the ordinates of those zeros by eigenangles of Haar-distributed…

Moment conjecture. For $k>2$, there is a positive constant $c_k$ such that

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 $J_{k,m}(T):=\int_1^T\Delta^k(t)|\zeta(1/2+it)|^{2m}\,dt$, with fixed integers $k\geq2$ and $m\geq1$. Ivić's hybrid-moment conjecture. There are a polynomial $Q_{m^2}(x)$ of de…

Ivić–Zhai conjecture. One has

Let the non-trivial zeros of the Riemann zeta function be written as $\rho=\tfrac12+i\gamma$. Assume the Riemann Hypothesis and that these zeros are simple. Ng's conjecture. As…

For the Riemann zeta function, let $K(\alpha)$ denote the normalized form factor associated with the pair correlation of its nontrivial zeros. Goldston–Montgomery GUE form-factor c…

The quasi-Riemann hypothesis. If this claim holds, then in the theorem's setting the auxiliary condition (b) on the coefficients is unnecessary. This is a conditional simplificatio…

Exponent pair hypothesis. One should have

Generalized zeta-derivative moment conjecture. The asymptotic relations stated in the source should hold, in particular