We consider an arithmetic function defined independently by John G. Thompson and Greg Simay, with particular attention to its mean value, its maximal size, and the analytic nature of its Dirichlet series generating function.

There are several formulas in classical prime number theory that are said to be “equivalent” to the Prime Number Theorem. For Beurling generalized numbers, not all such implications hold unconditionally. Here we investigate conditions under which the Beurling version of these relations do or do not hold.

Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an $n$ -point set $P$ , and a collection ${\mathcal{F}}=\{F_{1},\ldots ,F_{m}\}$ of subsets of $P$ , and our goal is color $P$ with two colors, red and blue, so that the maximum over the $F_{i}$ of the absolute difference between the number of red elements and the number of blue elements (the discrepancy) is minimized. Combinatorial discrepancy has many applications in mathematics and computer science, including constructions of uniformly distributed point sets, and lower bounds for data structures and private data analysis algorithms. We investigate the combinatorial discrepancy of geometrically defined systems, in which $P$ is an $n$ -point set in $d$ -dimensional space, and ${\mathcal{F}}$ is the collection of subsets of $P$ induced by dilations and translations of a fixed convex polytope $B$ . Such set systems include systems of sets induced by axis-aligned boxes, whose discrepancy is the subject of the well-known Tusnády problem. We prove new discrepancy upper and lower bounds for such set systems by extending the approach based on factorization norms previously used by the author, Matoušek, and Talwar. We also outline applications of our results to geometric discrepancy, data structure lower bounds, and differential privacy.

The discrepancy function measures the deviation of the empirical distribution of a point set in $[0,1]^{d}$ from the uniform distribution. In this paper, we study the classical discrepancy function with respect to the bounded mean oscillation and exponential Orlicz norms, as well as Sobolev, Besov and Triebel–Lizorkin norms with dominating mixed smoothness. We give sharp bounds for the discrepancy function under such norms with respect to infinite sequences.

Introduced in Schmidt and Summerer [Parametric geometry of numbers and applications. Acta Arith.140 (2009), 67–91], approximation exponents $\text{}\underline{\unicode[STIX]{x1D711}}_{i},\overline{\unicode[STIX]{x1D711}}_{i}$ , $(i=1,2,3)$ , attached to points $\boldsymbol{\unicode[STIX]{x1D709}}=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2})$ in $\mathbb{R}^{2}$ , give information on Diophantine approximation properties of these points. Laurent [Exponents of Diophantine approximation in dimension two. Canad. J. Math.61 (2009), 165–189] had described all possible quadruples $(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{3})$ arising in this way. Our emphasis here will be on $\text{}\underline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{2}$ and the construction of suitable “ $3$ -systems”.

The multiplicative group generated by a certain sequence of rationals is determined, settling a 30-year conjecture.

Recently, an analogue over $\mathbb{F}_{q}[T]$ of Landau’s theorem on sums of two squares was considered by Bary-Soroker, Smilansky and Wolf. They counted the number of monic polynomials in $\mathbb{F}_{q}[T]$ of degree $n$ of the form $A^{2}+TB^{2}$ , which we denote by $B(n,q)$ . They studied $B(n,q)$ in two limits: fixed $n$ and large $q$ ; and fixed $q$ and large $n$ . We generalize their result to the most general limit $q^{n}\rightarrow \infty$ . More precisely, we prove

$$\begin{eqnarray}B(n,q)\sim K_{q}\cdot \binom{n-\frac{1}{2}}{n}\cdot q^{n},\quad q^{n}\rightarrow \infty ,\end{eqnarray}$$

$K_{q}=1+O(1/q)$

We study the $p$ -adic variation of triangulations over $p$ -adic families of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules. In particular, we study certain canonical sub-filtrations of the pointwise triangulations and show that they extend to affinoid neighborhoods of crystalline points. This generalizes results of Kedlaya, Pottharst and Xiao and (independently) Liu in the case where one expects the entire triangulation to extend. We also study the ramification of weight parameters over natural $p$ -adic families.

We prove that if $k$ and $\ell$ are sufficiently large, then all the zeros of the weight $k+\ell$ cusp form $E_{k}(z)E_{\ell }(z)-E_{k+\ell }(z)$ in the standard fundamental domain lie on the boundary. We, moreover, find formulas for the number of zeros on the bottom arc with $|z|=1$ , and those on the sides with $Re(z)=\pm 1/2$ . One important ingredient of the proof is an approximation of the Eisenstein series in terms of the Jacobi theta function.

A generalization of Serre’s Conjecture asserts that if $F$ is a totally real field, then certain characteristic $p$ representations of Galois groups over $F$ arise from Hilbert modular forms. Moreover, it predicts the set of weights of such forms in terms of the local behaviour of the Galois representation at primes over  $p$ . This characterization of the weights, which is formulated using $p$ -adic Hodge theory, is known under mild technical hypotheses if $p>2$ . In this paper we give, under the assumption that $p$ is unramified in $F$ , a conjectural alternative description for the set of weights. Our approach is to use the Artin–Hasse exponential and local class field theory to construct bases for local Galois cohomology spaces in terms of which we identify subspaces that should correspond to ones defined using $p$ -adic Hodge theory. The resulting conjecture amounts to an explicit description of wild ramification in reductions of certain crystalline Galois representations. It enables the direct computation of the set of Serre weights of a Galois representation, which we illustrate with numerical examples. A proof of this conjecture has been announced by Calegari, Emerton, Gee and Mavrides.

We apply multigrade efficient congruencing to estimate Vinogradov’s integral of degree $k$ for moments of order $2s$ , establishing strongly diagonal behaviour for $1\leqslant s\leqslant \frac{1}{2}k(k+1)-\frac{1}{3}k+o(k)$ . In particular, as $k\rightarrow \infty$ , we confirm the main conjecture in Vinogradov’s mean value theorem for a proportion asymptotically approaching $100\%$ of the critical interval $1\leqslant s\leqslant \frac{1}{2}k(k+1)$ .

Let $c\unicode[STIX]{x1D719}_{k}(n)$ denote the number of $k$ -colored generalized Frobenius partitions of $n$ . Recently, new Ramanujan-type congruences associated with $c\unicode[STIX]{x1D719}_{4}(n)$ were discovered. In this article, we discuss two approaches in proving such congruences using the theory of modular forms. Our methods allow us to prove congruences such as $c\unicode[STIX]{x1D719}_{4}(14n+6)\equiv 0\;\text{mod}\;7$ and Seller’s congruence $c\unicode[STIX]{x1D719}_{4}(10n+6)\equiv 0\;\text{mod}\;5$ .

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field. Let $(\unicode[STIX]{x1D70B}_{n})_{n\geqslant 0}$ be a system of $p$ -power roots of a uniformizer $\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}_{0}$ of $K$ with $\unicode[STIX]{x1D70B}_{n+1}^{p}=\unicode[STIX]{x1D70B}_{n}$ , and define $G_{s}$ (resp.  $G_{\infty }$ ) the absolute Galois group of $K(\unicode[STIX]{x1D70B}_{s})$ (resp.  $K_{\infty }:=\bigcup _{n\geqslant 0}K(\unicode[STIX]{x1D70B}_{n})$ ). In this paper, we study $G_{s}$ -equivariantness properties of $G_{\infty }$ -equivariant homomorphisms between torsion crystalline representations.

Let $\widetilde{\text{Sp}}(2n)$ be the metaplectic covering of $\text{Sp}(2n)$ over a local field of characteristic zero. The core of the theory of endoscopy for $\widetilde{\text{Sp}}(2n)$ is the geometric transfer of orbital integrals to its elliptic endoscopic groups. The dual of this map, called the spectral transfer, is expected to yield endoscopic character relations which should reveal the internal structure of $L$-packets. As a first step, we characterize the image of the collective geometric transfer in the non-archimedean case, then reduce the spectral transfer to the case of cuspidal test functions by using a simple stable trace formula. In the archimedean case, we establish the character relations and determine the spectral transfer factors by rephrasing the works by Adams and Renard.

We determine the Galois representations inside the $\ell$-adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places. The main results generalize the previous works of Reimann and Kottwitz in this setting to arbitrary levels at $p$, and confirm the expected description of the cohomology due to Langlands and Kottwitz.

Let $B_{k,\ell }(n)$ denote the number of $(k,\ell )$ -regular bipartitions of $n$ . Employing both the theory of modular forms and some elementary methods, we systematically study the arithmetic properties of $B_{3,\ell }(n)$ and $B_{5,\ell }(n)$ . In particular, we confirm all the conjectures proposed by Dou [‘Congruences for (3,11)-regular bipartitions modulo 11’, Ramanujan J.40, 535–540].

Let $r\geq 2$ and $s\geq 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$ -ary expansion and the $s$ -ary expansion of an irrational real number, viewed as infinite words on $\{0,1,\ldots ,r-1\}$ and $\{0,1,\ldots ,s-1\}$ , and we show that this bound is best possible.

We determine the Galois representations inside the $\ell$-adic cohomology of some unitary Shimura varieties at split places where they admit uniformization by finite products of Drinfeld upper half spaces. Our main results confirm Langlands–Kottwitz’s description of the cohomology of Shimura varieties in new cases.

We introduce the Hurwitz-type spectral zeta functions for the quantum Rabi models, and give their meromorphic continuation to the whole complex plane with only one simple pole at $s=1$ . As an application, we give the Weyl law for the quantum Rabi models. As a byproduct, we also give a rationality of Rabi–Bernoulli polynomials introduced in this paper.

We define weight changing operators for automorphic forms on Grassmannians, that is, on orthogonal groups, and investigate their basic properties. We then evaluate their action on theta kernels, and prove that theta lifts of modular forms, in which the theta kernel involves polynomials of a special type, have some interesting differential properties.