11 problems

Let $B(R)$ be the disc of radius $R$, and let … be the number of lattice points in it. Hardy's conjecture. One has … This is the central conjecture of the Gauss circle problem. The…

Circle problem conjecture. One has

Lester–Wigman conjecture. The area correlations $\widetilde{C}_{k}$ tend to zero as the separation $k$ tends to infinity. The authors also established the Cesàro-mean identity…

Let … be the error term in counting lattice points in a disk. Gauss circle problem conjecture. The optimal order of $P_2(x)$ is $1/4$; equivalently, for every $\epsilon>0$, … Deter…

Let $\Delta(x)$ and $P(x)$ denote the error terms in the Dirichlet divisor problem and the Gauss circle problem, respectively, and write $\log_2 x=\log\log x$. For $X\geqslant 1$,…

Quarter-exponent conjecture. It is conjectured that $\theta_i=1/4$ for both the Dirichlet divisor problem and the Gauss circle problem. This is the classical conjectured optimal ex…

Let $r_0(n)$ denote the number of representations of $n$ as the sum of the squares of a positive and a non-negative integer, with representations corresponding to lattice points…

Let $r(n)$ denote the number of ways to write $n$ as a sum of two squares of integers, and let … It is known that $R(x)=\pi x+O(x^\theta)$ for some $\theta\in(\frac14,\frac12)$. Ci…

Quarter-exponent conjecture. In Gauss's Circle Problem and Dirichlet's Divisor Problem,

Let $r_k(m)$ count the integer $k$-tuples $(n_1,n_2,\ldots,n_k)$ satisfying $n_1^2+\cdots+n_k^2=m$, and let $S_k(n)=\sum_{0\leq m\leq n}r_k(m)$. Writing … where $V_k$ is the volume…

Quarter-exponent conjecture. It is conjectured that