We show that Hermite’s theorem fails for every integer $n$ of the form $3^{k_{1}}+3^{k_{2}}+3^{k_{3}}$ with integers $k_{1}>k_{2}>k_{3}\geqslant 0$. This confirms a conjecture of Brassil and Reichstein. We also obtain new results for the relative Hermite–Joubert problem over a finitely generated field of characteristic 0.

Darmon, Lauder, and Rotger conjectured that the relative tangent space of an eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of $p$-adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.

We answer a challenge posed in Booker [$L$-functions as distributions. Math. Ann. 363(1–2) (2015), 423–454, §1.3] by proving a version of Weil’s converse theorem [Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149–156] that assumes a functional equation for character twists but allows their root numbers to vary arbitrarily.

We generalize a method of Conrey and Ghosh [Simple zeros of the Ramanujan $\unicode[STIX]{x1D70F}$-Dirichlet series. Invent. Math. 94(2) (1988), 403–419] to prove quantitative estimates for simple zeros of modular form $L$-functions of arbitrary conductor.

We aim to re-prove a theorem conjectured by Gauss, namely there are exactly nine imaginary quadratic fields $\mathbf{Q}(\sqrt{-q})$ with class number one: specifically the list is $q\in \{3,4,7,8,11,19,43,67,163\}$. Our method initially follows an idea of Goldfeld, but rather than using an elliptic curve of analytic rank three (provided by the Gross–Zagier theorem), we instead use an elliptic curve of analytic rank two, where this $L$-function vanishing can be proven by modular symbols rather than a difficult height formula. It is already clear that Goldfeld’s work yields a constant lower bound for the class number by such means, but unfortunately it seems that even for the best choice of elliptic curve this numerical constant is less than 1, unless one can show non-trivial cancellation in the $L$-function coefficients restricted to values taken by quadratic forms. To show the latter, we consider a specific analytic rank-two elliptic curve with complex multiplication by $\mathbf{Q}(\sqrt{-1})$, and then by adapting a result of Hooley’s regarding equi-distrbution of roots of a quadratic polynomial to varying moduli, are able to show that there is indeed sufficient coefficient cancellation, giving an effective resolution of class number one. As we use various aspects of the principal form, our proof seems inapplicable for larger class numbers. We also comment on the possibility of using spectral techniques (following Templier and Tsimerman) to show the desired coefficient cancellation, though postpone the details of this to elsewhere.

We establish Diophantine inequalities for the fractional parts of generalized polynomials, in particular for sequences $\unicode[STIX]{x1D708}(n)=\lfloor n^{c}\rfloor +n^{k}$ with $c>1$ a non-integral real number and $k\in \mathbb{N}$, as well as for $\unicode[STIX]{x1D708}(p)$ where $p$ runs through all prime numbers. This is related to classical work of Heilbronn and to recent results of Bergelson et al.

In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let $\ell$ be a prime, $q$ a prime power and consider the ensemble ${\mathcal{H}}_{g,\ell }$ of $\ell$-cyclic covers of $\mathbb{P}_{\mathbb{F}_{q}}^{1}$ of genus $g$. We assume that $q\not \equiv 0,1~\text{mod}~\ell$. If $2g+2\ell -2\not \equiv 0~\text{mod}~(\ell -1)\operatorname{ord}_{\ell }(q)$, then ${\mathcal{H}}_{g,\ell }$ is empty. Otherwise, the number of rational points on a random curve in ${\mathcal{H}}_{g,\ell }$ distributes as $\sum _{i=1}^{q+1}X_{i}$ as $g\rightarrow \infty$, where $X_{1},\ldots ,X_{q+1}$ are independent and identically distributed random variables taking the values $0$ and $\ell$ with probabilities $(\ell -1)/\ell$ and $1/\ell$, respectively. The novelty of our result is that it works in the absence of a primitive $\ell$th root of unity, the presence of which was crucial in previous studies.

We generalize the work of Sarnak and Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik–Selberg conjecture.

In this paper various analytic techniques are combined in order to study the average of a product of a Hecke $L$-function and a symmetric square $L$-function at the central point in the weight aspect. The evaluation of the second main term relies on the theory of Maaß forms of half-integral weight and the Rankin–Selberg method. The error terms are bounded using the Liouville–Green approximation.

We prove that for every sufficiently large integer $n$, the polynomial $1+x+x^{2}/11+x^{3}/111+\cdots +x^{n}/111\ldots 1$ is irreducible over the rationals, where the coefficient of $x^{k}$ for $1\leqslant k\leqslant n$ is the reciprocal of the decimal number consisting of $k$ digits which are each $1$. Similar results following from the same techniques are discussed.

We discuss 1-factorizations of complete graphs that “match” a given Hadamard matrix. We prove the existence of these factorizations for two families of Hadamard matrices: Walsh matrices and certain Paley matrices.

We consider sequences of the form $(a_{n}\unicode[STIX]{x1D6FC})_{n}$ mod 1, where $\unicode[STIX]{x1D6FC}\in [0,1]$ and where $(a_{n})_{n}$ is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all $\unicode[STIX]{x1D6FC}$ in the sense of Lebesgue measure, we say that $(a_{n})_{n}$ has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of $(a_{n})_{n}$. Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterizes the metric pair correlation property in terms of the additive energy, similar to Khintchine’s criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence $(a_{n})_{n}$ having large additive energy which, however, maintains the metric pair correlation property.

In this paper, we improve the moment estimates for the gaps between numbers that can be represented as a sum of two squares of integers. We consider a certain sum of Bessel functions and prove the upper bound for its mean value. This bound provides estimates for the $\unicode[STIX]{x1D6FE}$th moments of gaps for all $\unicode[STIX]{x1D6FE}\leqslant 2$.

Let $X:=\mathbb{A}_{R}^{n}$ be the $n$-dimensional affine space over a discrete valuation ring $R$ with fraction field $K$. We prove that any pointed torsor $Y$ over $\mathbb{A}_{K}^{n}$ under the action of an affine finite-type group scheme can be extended to a torsor over $\mathbb{A}_{R}^{n}$ possibly after pulling $Y$ back over an automorphism of $\mathbb{A}_{K}^{n}$. The proof is effective. Other cases, including $X=\unicode[STIX]{x1D6FC}_{p,R}$, are also discussed.

We obtain a non-trivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short intervals, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However, the underlying technique is different and allows us to obtain non-trivial results starting from much shorter ranges.

We study the fine-scale $L^{2}$-mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain’s eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.

We examine correlations of the Möbius function over $\mathbb{F}_{q}[t]$ with linear or quadratic phases, that is, averages of the form 1

$$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f<n}\unicode[STIX]{x1D707}(f)\unicode[STIX]{x1D712}(Q(f))\end{eqnarray}$$

$\unicode[STIX]{x1D712}$

$\mathbb{F}_{q}$

$Q\in \mathbb{F}_{q}[x_{0},\ldots ,x_{n-1}]$

$x_{0},\ldots ,x_{n-1}$

$f=\sum _{i<n}x_{i}t^{i}$

$\unicode[STIX]{x1D707}$

$O_{\unicode[STIX]{x1D716}}(q^{(-1/4+\unicode[STIX]{x1D716})n})$

$\unicode[STIX]{x1D716}>0$

$Q$

$O(q^{-n^{c}})$

$c>0$

$Q$

$O(q^{-c^{\prime }n})$

$c^{\prime }>0$

$Q(f)$

$f^{2}$

In this article we obtain an explicit formula for certain Rankin–Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler–Zagier–Ibukiyama correspondence. The integral weight version of the main theorem was obtained by Katsurada and Kawamura. The result of the integral weight case is a product of an $L$-function and Riemann zeta functions, while the half-integral weight case is an infinite summation over negative fundamental discriminants with certain infinite products. To calculate an explicit formula for such Rankin–Selberg type Dirichlet series, we use a generalized Maass relation and adjoint maps of index-shift maps of Jacobi forms.

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of $t$-designs on such spaces are also considered (Theorem 2.3). In particular, it is shown that the optimal $t$-designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).

Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivić, Schwarz, Wirsing and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert’s result

$$\begin{eqnarray}\max _{n\leqslant x}\log d(n)=\frac{\log x}{\log \log x}(\log 2+o(1)).\end{eqnarray}$$

$\log f(f(n))$

$f$

$f$

$r_{2}$

$$\begin{eqnarray}\max _{n\leqslant x}\log r_{2}(r_{2}(n))=\frac{\sqrt{\log x}}{\log \log x}(c/\sqrt{2}+o(1))\end{eqnarray}$$

$c>0$