9 problems

Montgomery's large value conjecture. One should have

Let $T$ be sufficiently large and let $K=T^{1+\eta}$ with $\eta\in(0,1)$. For $k\in\mathbb{N}$, define the $k$-th divisor function by … Define … … where … for $n\in\mathbb{Z}^+$ an…

Harper's conjecture. For all $0\leq q\leq 1$ and any fixed $A>0$, for large prime $r$,

Let $M_{Dir}$ be the matrix associated with Dirichlet polynomials $D(t)=\sum_{n=N}^{2N}b_n e^{i t\log n}$, with $|b_n|\leq1$, and suppose $N\leq T\leq N^{O(1)}$. Montgomery's…

For $1/2<\sigma\leq 1$ and $\tau\geq 0$, let $\mathrm{LV}(\sigma,\tau)$ denote the large-values exponent for Dirichlet polynomials in the source: it measures the exponent governing…

Smooth-number approximation conjecture. There exists a constant $A>0$ such that, for any $1\le x\le T$ and $2T\le t\le 5T$, uniformly as $T\to\infty$,

Let $N\geq 1$, $T>0$, and let $B_T$ be any ball of radius $T$. For coefficients $\{b_n\}_{n=N+1}^{2N}\subset\mathbb{C}$, write … for the associated Dirichlet polynomial, and let…

Prime-polynomial symmetry conjecture. The positive and negative real parts, the real and imaginary parts, and the positive and negative imaginary parts have comparable maximal size…

Let $N\geq 2$ be an integer and let $a_n$ be complex numbers for $2\leq n\leq N$. Ramachandra's conjecture. For each $\delta>0$, there exists an $H>0$ such that … for all integers…