7 problems

Pointwise bound conjecture for ${\Cal Z}_2$. The exponent $\frac{4}{3}(1-\sigma)$ should be replaceable by $\frac{1}{2}-\sigma$, so that

Let $r\geq 1$ be fixed. For $h\sim H$, define the moment $M_{2r}(H,n)$ as in the source. Ideal moment conjecture. For every fixed $\delta>0$ and every $\varepsilon>0$, one expects…

Mean and second-moment conjecture. Computations support the conjecture that

Let $f$ be the modular form underlying the twisted $L$-function, let $X>0$, and let $\ell$ be a positive odd integer written as $\ell=\ell_1\ell_2^2$ with $\ell_1$ square-free. Def…

Let $\underline{k}=(k_1,k_2)$ with $k_1\ge k_2\ge3$, let $\mu_F(m)$ be the Dirichlet coefficients of the degree-$5$ standard $L$-function of $F$, and write…

Let $\Delta(t)$ denote the error term in the Dirichlet divisor problem. For a given natural number $k>1$, consider the $k$-th moment of this error term, $\int_0^T\Delta^k(t)\,dt$.…

Let … where $\gamma$ is Euler's constant, and let … where $P_4$ is a degree-four polynomial with leading coefficient $1/(2\pi^2)$. Conjectured bounds. For every $\varepsilon>0$, ……