213 problems

Let $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ be the Riemann zeta-function, and let $t$ be real. The notation $f(t)\ll g(t)$ means that $|f(t)|\leq Cg(t)$ for some constant $C$ in the r…

Gonek–Hejhal conjecture at $k=1$. One has

Let $\rho=\beta+i\gamma$ range over the non-trivial zeros of the Riemann zeta function, and define … Let $G$ denote Barnes' $G$-function, defined for $s\in\mathbb{C}$ by … For…

Soundararajan's conjecture. For $k\geq 2$ and $T\geq 10$,

Linear Independence Conjecture. This set is linearly independent over $\mathbb{Q}$. The conjecture is used in the paper's large-deviation arguments for the summatory functions unde…

Let $\zeta(s)$ be the Riemann zeta function, let $\rho=\beta+i\gamma$ range over its non-trivial zeros, and let $\zeta'(\rho)$ denote the derivative of $\zeta$ at $\rho$. Shanks' c…

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 $\psi_2(n)=\sum_{m+m'=n}\Lambda(m)\Lambda(m')$, where $\Lambda$ is the von Mangoldt function. Fujii's formula, under the Riemann Hypothesis, expresses the average Goldbach repr…

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 $\zeta(s)$ denote the Riemann zeta function, let $\gamma$ be Euler's constant, and let $C_1$ be a constant. Write $\log_2 T=\log(\log T)$ and $\log_3 T=\log(\log(\log T))$. Ref…

Let $N(\sigma,T)$ denote the number of nontrivial zeros $\rho=\beta+i\gamma$ of the Riemann zeta-function with $\beta\ge\sigma$ and $|\gamma|\le T$. Density Hypothesis. For all…

Let $T\to\infty$ and let $k>0$. Consider $t$ uniformly in $[T,2T]$, and let $a_k$ be the arithmetic factor from the zeta-function moment conjecture and…

Farmer's conjecture. The asymptotic formula above should remain valid with $\theta$ arbitrarily large.

Let $P$ be an algebraic differential expression in $\zeta$ whose coefficients are polynomials in the Euler gamma function $\Gamma$ and its derivatives; equivalently, $P$ is a polyn…

Let $\zeta(s)$ denote the Riemann zeta function, let $T\geq 1$, and write $\gamma$ for the Euler–Mascheroni constant. Littlewood's maximum-order conjecture. … This conjecture predi…

Let $\zeta(s)$ be the Riemann zeta-function and let its zeros be counted with their multiplicities. Simple zeros conjecture. All zeros of $\zeta(s)$ are simple. This is presented a…

Montgomery's conjecture. The non-trivial zeros satisfy the statistics of the eigenvalues of random hermitian matrices.

Let $\rho=\beta+i\u03b3$ and $\rho'=\beta'+i\u03b3'$ denote non-trivial zeros of $\zeta(s)$ and $\zeta'(s)$, respectively, and assume the Riemann Hypothesis. Let $\gamma^+$ be the…

Soundararajan's conjecture. The equality $\lambda=0$ holds if and only if $\lambda'=0$. This conjecture relates unusually small gaps between consecutive zeta zeros to zeros of…

Assume the Riemann hypothesis. Let $\zeta(s)$ be the Riemann zeta function, write $s=\sigma+it$, let $c>0$, and let $\sigma_1<1$. Let $\gamma$ and $\gamma^+$ denote consecutive ord…

Modified quarter-exponent conjecture. It is conjectured that

Equality-of-exponents conjecture. A weaker conjecture is

The quarter-exponent conjecture. One conjectures that

Let $\zeta(s)$ denote the Riemann zeta function. An algebraic differential equation with rational coefficients is an equation of the form … where $n$ is a nonnegative integer and…

Conjecture for $\mathcal Z_2$. For $1/2<\sigma\leq 1$ and $t\geq t_0>0$, one has