Number Theory Papers of the Week

We construct infinite $B_h$ sets that repeatedly attain the density scale of finite $B_h$ sets.

We use an automorphic L-function over one number field as a probe for the functorial structure of another automorphic L-function over a possibly different field.

We therefore adjoin a specified cuspidal L-function as a probe and study the resulting raw mixed correlations. Assuming a factorization into cuspidal L-functions over Q, we show that a mixed pair statistic detects the probe's occurrence and recovers its multiplicity.

We prove the complementary-class, short-sum and algebraic comparison inequalities directly, and obtain the endpoint $B_*-1/160000=0.87446205769353\ldots$, where $B_*$ is Liu's algebraic boundary.

We prove a quantitative non-covering theorem for finite nilpotent groups.

We construct a non-admissible smooth irreducible representation of $\mathrm{GL}_2(F)$ over $\overline{\mathbb{F}}_p$.

We offer a simple explanation in terms of self-duality under asymmetric orbifolds.

Using Pan's approximate factorization theorem, we give a deterministic algorithm that finds all distinct integer roots of $f$ in $\widetilde O(nb)$ bit operations.

We prove that the shifts $\zeta(s+id_j\tau)$, $1\le j\le n$, are jointly universal for any nonzero real parameters $d_j$ with pairwise distinct absolute values.

\end{align*} They asked whether there exists an integer $m>2$ for which $a'_m(n)$ is eventually positive, and posed the same question for $b'_m(n)$. In this paper, we resolve these two questions.

We investigate the structure of locally analytic representations of $p$-adic groups, and establish a sufficient condition for the irreducibility of locally analytic vectors in certain Banach representations of $D^\times$, where $D$ is the quaternion division algebra over $\mathbb{Q}_p$.

We prove twisting-free converse theorems for $\Gamma_0(32)$ and $\Gamma_0(36)$ under the setting of Conrey--Farmer, namely assuming a single functional equation and a suitable Euler product, with particular shapes at ramified primes.

We give a counting formula for the coefficients $a\in\F_{p^h}$ for which $x^{2n}+ax^n+1$ is irreducible in $\F_{p^h}[x;\sigma]$, for arbitrary positive integers $n$ and $h$, where $\sigma(a)=a^p$ and $p$ is an arbitrary prime power.

We establish p-adic formulas for the eigenvalues of Frobenius in the case of the family of hyperkloosterman sums.

We determine the order of magnitude of \[ \mathbb{E} \Bigg[ \bigg| \frac{1}{\sqrt{x}}\sum_{n \le x} X^e(n) \bigg|^{2 q} \Bigg]\quad \text{ and }\quad \mathbb{E} \Bigg[ \bigg|\sum_{n \le x} \frac{X(n)}{\sqrt{n}} \bigg|^{2 q} \Bigg] \] for $q\in [0,1]$, showing that a transition from better-than-square-root cancellation to growth like a power of $\log x$ occurs when $q$ crosses $1/2$.

The ensemble average of Narain conformal field theories over their moduli space is a real-analytic Eisenstein series, whose Poincaré expansion can be interpreted as a sum over lens spaces weighted by the partition functions of Abelian Chern-Simons theory. In this paper we extend this correspondence from lens spaces to Seifert manifolds, which gives an example of a restricted sum over topologies of the kind that arises in three-dimensional quantum gravity.

In this paper, we complete the classification of good rational functions of degree up to $6$ in characteristic $2$.

In this paper, we provide an upper bound on the number of a special system of Roth-type Diophantine inequalities over proper adelic curves.

In the second paper in this series, we will prove that \[ \limsup_{X \to \infty} \frac{\log \mathcal{I}_{\lambda}(X)} {\log X} \geqslant\frac{1}{3}.