We study the moments $M_k(T;\,\alpha) = \int_T^{2T} |\zeta(s,\alpha)|^{2k}\,dt$ of the Hurwitz zeta function $\zeta(s,\alpha)$ on the critical line, $s = 1/2 + it$ with a rational shift $\alpha \in \mathbb{Q}$. We conjecture, in analogy with the Riemann zeta function, that $M_k(T;\,\alpha) \sim c_k(\alpha) T (\!\log T)^{k^2}$. Using heuristics from analytic number theory and random matrix theory, we conjecturally compute $c_k(\alpha)$. In the process, we investigate moments of products of Dirichlet L-functions on the critical line. We prove some of our conjectures for the cases $k = 1,2$.

We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x \in (0,1]$ with the following property is comeager: for all integers $b\ge 2$ and $k\ge 1$, the sequence of vectors made by the frequencies of all possibile strings of length k in the b-adic representation of x has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen (2004) in this Journal. We provide analogues in the context of analytic P-ideals and regular matrices.

A superelliptic curve over a discrete valuation ring $\mathscr{O}$ of residual characteristic p is a curve given by an equation $\mathscr{C}\;:\; y^n=\,f(x)$, with $\textrm{Disc}(\,f)\neq 0$. The purpose of this article is to describe the Galois representation attached to such a curve under the hypothesis that f(x) has all its roots in the fraction field of $\mathscr{O}$ and that $p \nmid n$. Our results are inspired on the algorithm given in Bouw and WewersGlasg (Math. J. 59(1) (2017), 77–108.) but our description is given in terms of a cluster picture as defined in Dokchitser et al. (Algebraic curves and their applications, Contemporary Mathematics, vol. 724 (American Mathematical Society, Providence, RI, 2019), 73–135.).

For a positive integer $r\geq 2$, a natural number n is r-free if there is no prime p such that $p^r\mid n$. Asymptotic formulae for the distribution of r-free integers in the floor function set $S(x):=\{\lfloor x/ n \rfloor :1\leq n\leq x\}$ are derived. The first formula uses an estimate for elements of $S(x)$ belonging to arithmetic progressions. The other, more refined, formula makes use of an exponent pair and the Riemann hypothesis.

In their renowned paper (2011, Inventiones Mathematicae 184, 591–627), I. Vollaard and T. Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport–Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat–Tits stratum, and a Deligne–Lusztig variety which is not of classical type. In this paper, we describe the $\ell $-adic cohomology groups over $\overline {{\mathbb Q}_{\ell }}$ of these Deligne–Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated with the Ekedahl–Oort stratification of a closed Bruhat–Tits stratum, which translates into a stratification by Coxeter varieties whose cohomology is known. Eventually, we find out that the irreducible representations of the finite unitary group which appear inside the cohomology contribute to only two different unipotent Harish-Chandra series, one of them belonging to the principal series.

Open conjectures state that, for every $x\in [0,1]$, the orbit $(x_n)_{n=1}^\infty $ of the mean-median recursion

$$ \begin{align*}x_{n+1}=(n+1)\cdot\operatorname{\mathrm{median}}(x_1,\ldots,x_{n})-(x_1+\cdots+x_n),\quad n\geqslant 3,\end{align*} $$

with initial data $(x_1,x_2,x_3)=(0,x,1)$, is eventually constant, and that its transit time and limit functions (of x) are unbounded and continuous, respectively. In this paper, we prove that for the slightly modified recursion

$$ \begin{align*}x_{n+1}=n\cdot\operatorname{\mathrm{median}}(x_1,\ldots,x_{n})-(x_1+\cdots+x_n),\quad n\geqslant 3,\end{align*} $$

first suggested by Akiyama, the transit time function is unbounded but the limit function is discontinuous.

Let $[t]$ be the integral part of the real number t and let $\mathbb {1}_{{\mathbb P}}$ be the characteristic function of the primes. Denote by $\pi _{\mathcal {S}}(x)$ the number of primes in the floor function set $\mathcal {S}(x) := \{[{x}/{n}] : 1\leqslant n\leqslant x\}$ and by $S_{\mathbb {1}_{{\mathbb P}}}(x)$ the number of primes in the sequence $\{[{x}/{n}]\}_{n\geqslant 1}$. Improving a result of Heyman [‘Primes in floor function sets’, Integers 22 (2022), Article no. A59], we show

$$ \begin{align*} \pi_{\mathcal{S}}(x) = \int_2^{\sqrt{x}} \frac{d t}{\log t} + \int_2^{\sqrt{x}} \frac{d t}{\log(x/t)} + O(\sqrt{x}\,\mathrm{e}^{-c(\log x)^{3/5}(\log\log x)^{-1/5}}) \quad\mbox{and}\quad S_{\mathbb{1}_{{\mathbb P}}}(x) = C_{\mathbb{1}_{{\mathbb P}}} x + O_{\varepsilon}(x^{9/19+\varepsilon}) \end{align*} $$

for $x\to \infty $, where $C_{\mathbb {1}_{{\mathbb P}}} := \sum _{p} {1}/{p(p+1)}$, $c>0$ is a positive constant and $\varepsilon $ is an arbitrarily small positive number.

We investigate a novel geometric Iwasawa theory for ${\mathbf Z}_p$-extensions of function fields over a perfect field k of characteristic $p>0$ by replacing the usual study of p-torsion in class groups with the study of p-torsion class group schemes. That is, if $\cdots \to X_2 \to X_1 \to X_0$ is the tower of curves over k associated with a ${\mathbf Z}_p$-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of the p-torsion group scheme in the Jacobian of $X_n$ as $n\rightarrow \infty $. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of $X_n$ equipped with natural actions of Frobenius and of the Cartier operator V. We formulate and test a number of conjectures which predict striking regularity in the $k[V]$-module structure of the space $M_n:=H^0(X_n, \Omega ^1_{X_n/k})$ of global regular differential forms as $n\rightarrow \infty .$ For example, for each tower in a basic class of ${\mathbf Z}_p$-towers, we conjecture that the dimension of the kernel of $V^r$ on $M_n$ is given by $a_r p^{2n} + \lambda _r n + c_r(n)$ for all n sufficiently large, where $a_r, \lambda _r$ are rational constants and $c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$ is a periodic function, depending on r and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on ${\mathbf Z}_p$-towers of curves, and we prove our conjectures in the case $p=2$ and $r=1$.

We introduce interpolated multiple Hurwitz polylogs and interpolated multiple Hurwitz zeta values. In addition, we discuss the generating functions for the sum of the polylogs/zeta values of fixed weight, depth, and all heights. The functions are expressed in terms of generalized hypergeometric functions. Compared with the pioneering results of Ohno and Zagier on the generating function, our setup generalizes the results in three directions, namely, at general heights, with a t-interpolation, and as a Hurwitz type. As an application, by fixing the Hurwitz parameter to rational numbers, the generating functions for multiple zeta values with level are given.

We prove Fermat’s Last Theorem over $\mathbb {Q}(\sqrt {5})$ and $\mathbb {Q}(\sqrt {17})$ for prime exponents $p \ge 5$ in certain congruence classes modulo $48$ by using a combination of the modular method and Brauer–Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of $\mathbb {Q}(\sqrt {5})$ is a generalization to a real quadratic base field of the one used by Chen and Siksek. For the case of $\mathbb {Q}(\sqrt {17})$, this is insufficient, and we generalize a reciprocity constraint of Bennett, Chen, Dahmen, and Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.

We settle a part of the conjecture by Bandini and Valentino [‘On the structure and slopes of Drinfeld cusp forms’, Exp. Math. 31(2) (2022), 637–651] for $S_{k,l}(\Gamma _0(T))$ when $\mathrm {dim}\ S_{k,l}(\mathrm {GL}_2(A))\leq 2$. We frame and check the conjecture for primes $\mathfrak {p}$ and higher levels $\mathfrak {p}\mathfrak {m}$, and show that a part of the conjecture for level $\mathfrak {p} \mathfrak {m}$ does not hold if $\mathfrak {m}\ne A$ and $(k,l)=(2,1)$.

Motivated by the desire to understand the geometry of the basic loci in the reduction of Shimura varieties, we study their “group-theoretic models”—generalized affine Deligne–Lusztig varieties—in cases where they have a particularly nice description. Continuing the work of Görtz and He (2015, Cambridge Journal of Mathematics 3, 323–353) and Görtz, He, and Nie (2019, Peking Mathematical Journal 2, 99–154), we single out the class of cases of Coxeter type, give a characterization in terms of the dimension, and obtain a complete classification. We also discuss known, new, and open cases from the point of view of Shimura varieties/Rapoport–Zink spaces.

A partition $\lambda $ of n is said to be nearly self-conjugate if the Ferrers graph of $\lambda $ and its transpose have exactly $n-1$ cells in common. The generating function of the number of such partitions was first conjectured by Campbell and recently confirmed by Campbell and Chern (‘Nearly self-conjugate integer partitions’, submitted for publication). We present a simple and direct analytic proof and a combinatorial proof of an equivalent statement.

Let a, b, c be fixed coprime positive integers with $\min \{ a,b,c \}>1$. Let $N(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y = c^z$. We show that if $(a,b,c)$ is a triple of distinct primes for which $N(a,b,c)>1$ and $(a,b,c)$ is not one of the six known such triples, then $c>10^{18}$, and there are exactly two solutions $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$ with $2 \mid x_1$, $2 \mid y_1$, $z_1=1$, $2 \nmid y_2$, $z_2>1$, and, taking $a<b$, we must have $a=2$, $b \equiv 1 \bmod 12$, $c \equiv 5\, \mod 12$, with $(a,b,c)$ satisfying further strong restrictions. These results support a conjecture put forward by Scott and Styer [‘Number of solutions to $a^x + b^y = c^z$’, Publ. Math. Debrecen 88 (2016), 131–138].

We study Ohno–Zagier type relations for multiple t-values and multiple t-star values. We represent the generating function of sums of multiple t-(star) values with fixed weight, depth and height in terms of the generalised hypergeometric function $\,_3F_2$. As applications, we get a formula for the generating function of sums of multiple t-(star) values of maximal height and a weighted sum formula for sums of multiple t-(star) values with fixed weight and depth.

In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type $\mathbf {A}_1+\mathbf {A}_3$ and prove an analogue of Manin’s conjecture for integral points with respect to its singularities and its lines.

We prove several finite product-sum identities involving the q-binomial coefficient, one of which is used to prove an amazing identity of Gauss. We then use this identity to evaluate certain quadratic Gauss sums and, together with known properties of quadratic Gauss sums, we prove the quadratic reciprocity law for the Jacobi symbol. We end our article with a new proof of Jenkins’ lemma, a lemma analogous to Gauss’ lemma. This article aims to show that Gauss’ amazing identity and the properties of quadratic Gauss sums are sufficient to establish the quadratic reciprocity law for the Jacobi symbol.

For each prime p, we show that there exist geometrically simple abelian varieties A over ${\mathbb Q}$ with . Specifically, for any prime $N\equiv 1 \ \pmod p$, let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point P of order p, and let $B = A_f/\langle P \rangle $. Then the number of positive integers $d \leq X$ with is $ \gg X/\log X$, where $\widehat B_d$ is the dual of the dth quadratic twist of B. We prove this more generally for abelian varieties of $\operatorname {\mathrm {GL}}_2$-type with a p-isogeny satisfying a mild technical condition. In the special case of elliptic curves, we give stronger results, including many examples where for an explicit positive proportion of integers d.

We construct a geometrico-symbolic version of the natural extension of the random $\beta $-transformation introduced by Dajani and Kraaikamp [Random $\beta $-expansions. Ergod. Th. & Dynam. Sys. 23(2) (2003) 461–479]. This construction provides a new proof of the existence of a unique absolutely continuous invariant probability measure for the random $\beta $-transformation, and an expression for its density. We then prove that this natural extension is a Bernoulli automorphism, generalizing to the random case the result of Smorodinsky [$\beta $-automorphisms are Bernoulli shifts. Acta Math. Acad. Sci. Hungar. 24 (1973), 273–278] about the greedy transformation.