20 problems

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

Prime-perturbation irreducibility conjecture. The Dirichlet polynomial

Turán-type perturbation conjecture. There is an absolute constant $C>0$ such that, for every such $f$, there is a bounded perturbation $g$ satisfying

Hilbert irreducibility conjecture. The polynomial $f$ remains irreducible for infinitely many integer specializations of $s$.

Bouniakovsky-type conjecture. A primitive, irreducible Dirichlet polynomial $f(s)$ with positive leading coefficient such that the set of values $f(\mathbb{Z}_{-})$ has no common d…

Let $P(n)$ denote the largest prime factor of $n$, let $f$ be a Steinhaus random multiplicative function, and fix $A>0$ and a sufficiently small $\varepsilon>0$. For $T>0$, exclude…

Let $M_{Dir}$ be the Dirichlet-polynomial matrix. Suppose $1<\alpha<2$ and $T=N^\alpha$. Let $W$ be a large-value set at threshold $N^\sigma$. Dirichlet sum-of-squares barrier conj…

Dirichlet majorant square-root conjecture. For $1\leq|t|\leq T$,

Let $M_{Dir}$ be the matrix associated with Dirichlet polynomials, and let $M_{Dir}M_{Dir}^*$ be its Gram matrix. Off-diagonal square-root conjecture. Every off-diagonal entry of…

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…

Higher-swap conjecture. In general, the $\ell$-swap terms should appear for $\eta\geq\ell$.

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 $\mathcal M_k(\alpha)$ be the normalized long-Dirichlet-polynomial mean square and let $\mathcal C_k(\alpha)$ be the normalized divisor variance in short intervals. Basor–Ge–Ru…

For $0\leq r\leq k$, define … where $d\boldsymbol w=dw_1\cdots dw_k$, and $C_i(r)$ is the positively oriented circle of radius $1/2$ centered at $-1$ for $i\leq r$ and at $0$ other…

Let $\mathcal M_3(\alpha)$ denote the normalized mean-square quantity defined in the paper, with $\alpha>0$. The cubic shifted-moment conjecture. … The formula predicts a continuou…

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…

For $1\leq p<q<\infty$ and $m\in\mathbb{N}$, let $H^{q,p}_{m}$ denote the comparison constant between the relevant $P$-norms of $m$-homogeneous Dirichlet polynomials. Optimal compa…

Let $A_N(s)=\sum_{n=1}^N\frac{a_n}{n^s}$ range over all Dirichlet polynomials of length $N$, and define … The Nyman–Beurling–Báez-Duarte criterion. The Riemann hypothesis is true i…

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…