1,798 problems

Let $\zeta(s)=\sum_{n\ge1}n^{-s}$ be the Riemann zeta function, continued meromorphically to $\mathbb{C}$. Its nontrivial zeros are the zeros $\rho=\beta+i\gamma$ in the critical s…

Let $r_2'(N)$ denote the number of representations of $N$ as a sum of two primes, and let $\mathfrak{S}(N)$ be the singular series … For even integers $N$, let $r_2(N)$ denote the…

Let $1/2<\sigma<1$ be fixed. Montgomery's lower bound concerns the maximum of $\log|\zeta(\sigma+it)|$ for $t\in[T,2T]$ as $T\to\infty$: … where $c(\sigma)>0$. Montgomery's conject…

Let $\chi$ be a primitive quadratic Dirichlet character, and let $L(s,\chi)$ denote its Dirichlet $L$-function. Chowla's conjecture. … The conjecture asserts non-vanishing at the c…

Let $L(s)$ range over the several families of $L$-functions considered by Keating and Snaith, evaluated at the central point $s=1/2$. Let $\mathrm{SO}(2N)$ and $\mathrm{USp}(2N)$ d…

Let $\mathfrak{L}_f(s)=\sum_{n=1}^{\infty}f(n)n^{-s}$ be a member of the Selberg class $\mathcal{S}$, with coefficients $f(n)$. Ramanujan's conjecture. One has…

Selberg's eigenvalue conjecture. For every $j$, $\theta_j=0$.

For $x>0$, let … be the prime-counting function. Gauss's conjecture. As $x\to\infty$, … The conjecture anticipated the Prime Number Theorem, proved independently by Hadamard and de…

Let $0<\theta<1$ and $C>0$ be fixed. Let $\Lambda(n)$ denote the von Mangoldt function and $\varphi(n)$ the Euler totient function. Elliott–Halberstam conjecture. For every $A>0$ a…

Let $p_n$ denote the $n$th prime. Cramér's conjecture. … This is a central conjecture on maximal prime gaps. The source notes that Granville challenged the constant $1$ in light of…

Let $E$ be an elliptic curve and let $a_p$ denote its Frobenius trace at a good prime $p$. For a fixed integer $t$, Lang–Trotter conjecture. … The conjecture predicts that primes w…

Strict monotonicity conjecture. The function $\boldsymbol D$ is strictly decreasing on $(0,1)$; in particular, $\boldsymbol\rho_0$ is its unique real zero.

Let $K/\mathbb{Q}$ be a finite Galois extension, and let $\zeta_K(s)$ be its Dedekind zeta function. Its non-trivial zeros are the zeros other than the trivial zeros arising from t…

Let $P\in \mathbb{Z}[t]$ be irreducible, and let $\mathfrak S_P$ denote the singular series … An integer with exactly two distinct prime divisors, called an $E_2$ integer, has the…

Let $n$ be an integer with $n\ge 1$. A prime number is an integer greater than $1$ with no positive divisors other than $1$ and itself. Legendre's conjecture. For each integer…

For a prime $q$, let $n(q)$ denote the least natural number that is not a quadratic residue modulo $q$. Vinogradov's conjecture. For every fixed $\varepsilon>0$, … This conjecture…

Let $L/K$ be a Galois extension of number fields with Galois group $G=\operatorname{Gal}(L/K)$. Let $\rho:G\to\operatorname{GL}(V)$ be an irreducible representation with character…

Let $\Delta(x)$ be defined by $\sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+\Delta(x)$, where $d(n)$ is the divisor function and $\gamma$ is Euler's constant. Dirichlet divisor problem…

Hilbert–Pólya-type conjecture. The nontrivial zeros of $\zeta(s)$ should be realized as eigenvalues of a self-adjoint operator; the coherence Hamiltonian provides an analogue whose…

Let $A_N(\gamma)$ be the oscillatory phase average associated with the centred layer moment, let $X$ be the relevant scale, and let $R$ be the corresponding correlation quantity. S…

Bunyakovsky's conjecture. The value $f(n)$ should be prime for infinitely many positive integers $n$.

Heath-Brown's conjecture. One has

Let … where $\mu$ is the Möbius function. Mertens' conjecture. The inequality … should hold for all $x>1$. This conjecture was based on Mertens' hand computation of $M(n)$ through…

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's conjecture. The normalized absolute value of the Mertens function has bounded limit superior: