51 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…

Ivić's mean-square conjecture.

Friedlander–Iwaniec Conjecture 2. For every $\varepsilon>0$,

Let … For any $0<\varepsilon<1/4$ and $x^\varepsilon\leqslant U\leqslant x^{1/2-\varepsilon}$, Jutila's short-interval conjecture. … This predicts square-root cancellation in the d…

Modified quarter-exponent conjecture. It is conjectured that

Equality-of-exponents conjecture. A weaker conjecture is

The quarter-exponent conjecture. One conjectures that

Quarter-power bounds conjecture. It is conjectured that

Generalized divisor-problem conjecture. One conjectures that

Let $\Delta_k(x)$ denote the error term in the generalized divisor problem, and define … For $k=2$, one traditionally conjectures that $\alpha_2=1/4$. This conjecture predicts the…

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 … be the summatory function of the ternary divisor function, where $d_3(n)$ counts the representations of $n$ as a product of three positive integers. Write … where…

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 $X\geqslant 1$,…

Let $\tau(n)$ denote the number of positive divisors of the integer $n\geqslant 1$. For $x\geqslant 1$, write … where $gamma$ is the Euler–Mascheroni constant and $Delta(x)$ is the…

Let $d(n)$ be the number of positive divisors of $n$, and define … where $\gamma$ is the Euler--Mascheroni constant and $\Delta(x)$ is the error term. Hardy's conjecture. The expon…

The pointwise bound conjecture. For every $\varepsilon>0$,

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…

Soundararajan's conjecture. The power $3/4(2^{4/3}-1)$ of $\log_2 x$ in this lower bound is optimal. The conjecture concerns the sharpness of the secondary logarithmic factor in th…

Let $d(n)$ denote the number of positive divisors of $n$, and define the divisor-problem error term … where $\gamma$ is Euler's constant. Let $\alpha$ be the infimum of the exponen…

Divisor-problem exponent conjecture. The admissible exponent $\theta=\frac{131}{416}$ can be reduced to $\frac14$.

Let $\Delta(x)$ be the remainder term in the summatory divisor problem, … where $d(n)$ is the divisor function, $\gamma$ is the Euler–Mascheroni constant, and $\log_k$ denotes the…

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

Aymone's Omega conjecture. The partial sums satisfy

For each integer $k\ge 2$, let $\Delta_k(x)$ be the remainder term in the generalised divisor problem, and define the associated exponents $\alpha_k$ and $\beta_k$ as in the source…