17 problems

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…

Hardy–Landau conjecture. Both error terms satisfy

Quarter-exponent conjecture. The exponent $1/4$ is the optimal exponent in both classical problems, equivalently, $1/4$ is the optimal remainder exponent for $\mathcal F(n)$ separa…

Let $k\geq2$ and $2\leq d_1<\cdots<d_k$ be fixed integers. For each $j$, let $\mathfrak{L}_{f_j}(s)\in\mathcal{S}_{\mathrm{real}}$ have degree $d_j\geq2$, and let $E_{f_j}(x)$ be i…

Let $S^{(1)}(x)$, $S^{(2)}(x)$, $C^{(1)}(x)$, and $C^{(2)}(x)$ be the subgroup-counting functions collected in $\mathcal{F}[x]$, and let … where $f(x)\in\mathcal{F}[x]$ and the…

Optimal error-term conjecture. The estimate

Square-root error conjecture. For fixed $t$, $m$, and $I\subset[-1,1]$,

Let $E$ be an elliptic curve without complex multiplication, and let $E(x,t,m,I)$ denote the error in the corresponding Chebotarev–Sato–Tate counting formula, with fixed $t$, $m$,…

Let $I_3(T)=\int_0^T|\zeta(\tfrac12+it)|^6\,dt$. Ivic's error-term conjecture. When all shifts vanish, the error term in the sixth-moment asymptotic is … for every $\varepsilon>0$.…

Let $I_3(T)=\int_0^T|\zeta(\tfrac12+it)|^6\,dt$. Motohashi's error-term conjecture. When all shifts vanish, the error term in the sixth-moment asymptotic is … for every fixed…

Let $D_k(x,h)=\sum_{n\le x}\tau_k(n)\tau_k(n+h)$, and let $E_k(x,h)$ denote the error term in the additive divisor conjecture. Vinogradov's conjecture. For fixed $h$, … This gives…

Conjectured omega bound. For every $\delta>0$,

Let $\Delta(x)$ be the error term in the asymptotic formula for partial sums of the square-free divisors, defined by … where $\omega(n)$ denotes the number of distinct prime diviso…

Signed error-term independence conjecture. Theorem 3.1 should remain valid for these coefficients and dilates:

Let $a$, $b$, $c$, and $k$ be positive integers such that $1\leq a\leq b$, $a<c<2(a+b)$, $c\ne b$, and $(a,b,c)=1$. Define the arithmetic function $f_k(a,b;c;n)$ by … and its summa…

Let $1\leq a\leq b$ be fixed integers with $(a,b)=1$, and define … For $a\ne b$, let … If $a=b$, take the appropriate limiting form. The generalized divisor-problem conjecture. For…

Pointwise bound conjecture. It is conjectured that