In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math. 163 (2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

This article considers the positive integers $N$ for which ${\it\zeta}_{N}(s)=\sum _{n=1}^{N}n^{-s}$ has zeroes in the half-plane $\Re (s)>1$ . Building on earlier results, we show that there are no zeroes for $1\leqslant N\leqslant 18$ and for $N=20,21,28$ . For all other $N$ there are infinitely many such zeroes.

We provide a comprehensive study of the function $h=h(q)$ defined by

$$\begin{eqnarray}h=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{12j-1})(1-q^{12j-11})}{(1-q^{12j-5})(1-q^{12j-7})}\end{eqnarray}$$

$k=k(q)$

$$\begin{eqnarray}k=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{10j-1})(1-q^{10j-2})(1-q^{10j-8})(1-q^{10j-9})}{(1-q^{10j-3})(1-q^{10j-4})(1-q^{10j-6})(1-q^{10j-7})}.\end{eqnarray}$$

We show that Ribet sections are the only obstruction to the validity of the relative Manin–Mumford conjecture for one-dimensional families of semi-abelian surfaces. Applications include special cases of the Zilber–Pink conjecture for curves in a mixed Shimura variety of dimension 4, as well as the study of polynomial Pell equations with non-separable discriminants.

We characterise number fields without a unit primitive element, and we exhibit some families of such fields with low degree. Also, we prove that a noncyclotomic totally complex number field $K$, with degree $2d$ where $d$ is odd, and having a unit primitive element, can be generated by a reciprocal integer if and only if $K$ is not CM and the Galois group of the normal closure of $K$ is contained in the hyperoctahedral group $B_{d}$.

Let $b_{\ell }(n)$ denote the number of $\ell$-regular partitions of $n$. In this paper we establish a formula for $b_{13}(3n+1)$ modulo $3$ and use this to find exact criteria for the $3$-divisibility of $b_{13}(3n+1)$ and $b_{13}(3n)$. We also give analogous criteria for $b_{7}(3n)$ and $b_{7}(3n+2)$.

For any positive integer $M$ we show that there are infinitely many real quadratic fields that do not admit $M$ -ary universal quadratic forms (without any restriction on the parity of their cross coefficients).

We study transcendence properties of certain infinite products of cyclotomic polynomials. In particular, we determine all cases in which the product is hypertranscendental. We then use various results from Mahler’s transcendence method to obtain algebraic independence results on such functions and their values.

For $n\in \mathbb{Z}$ and $A\subseteq \mathbb{Z}$ , define $r_{A}(n)$ and ${\it\delta}_{A}(n)$ by $r_{A}(n)=\#\{(a_{1},a_{2})\in A^{2}:n=a_{1}+a_{2},a_{1}\leq a_{2}\}$ and ${\it\delta}_{A}(n)=\#\{(a_{1},a_{2})\in A^{2}:n=a_{1}-a_{2}\}$ . We call $A$ a unique representation bi-basis if $r_{A}(n)=1$ for all $n\in \mathbb{Z}$ and ${\it\delta}_{A}(n)=1$ for all $n\in \mathbb{Z}\setminus \{0\}$ . In this paper, we prove that there exists a unique representation bi-basis $A$ such that $\limsup _{x\rightarrow \infty }A(-x,x)/\sqrt{x}\geq 1/\sqrt{2}$ .

We show that, by taking normalizations over certain auxiliary good reduction integral models, one obtains integral models of toroidal and minimal compactifications of PEL-type Shimura varieties which enjoy many features of the good reduction theory studied as in the earlier works of Faltings and Chai’s and the author’s. We treat all PEL-type cases uniformly, with no assumption on the level, ramifications, and residue characteristics involved.

We show that if a finite, large enough subset $A$ of an arbitrary abelian group $G$ satisfies the small doubling condition $|A+A|\leqslant (\log |A|)^{1-{\it\varepsilon}}|A|$ , then $A$ must contain a three-term arithmetic progression whose terms are not all equal, and $A+A$ must contain an arithmetic progression or a coset of a subgroup, either of which is of size at least $\exp [c(\log |A|)^{{\it\delta}}]$ . This extends analogous results obtained by Sanders, and by Croot, Łaba and Sisask in the case where $G=\mathbb{Z}$ .

For a half-integral weight modular form $f=\sum _{n=1}^{\infty }a_{f}(n)n^{(k-1)/2}q^{n}$ of weight $k=\ell +1/2$ on $\unicode[STIX]{x1D6E4}_{0}(4)$ such that $a_{f}(n)$ ( $n\in \mathbb{N}$ ) are real, we prove for a fixed suitable natural number $r$ that $a_{f}(n)$ changes sign infinitely often as $n$ varies over numbers having at most $r$ prime factors, assuming the analog of the Ramanujan conjecture for Fourier coefficients of half-integral weight forms.

Let $C\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ be a cubic form. Assume that $C$ splits into four forms. Then $C(x_{1},\ldots ,x_{n})=0$ has a non-trivial integer solution provided that $n\geqslant 10$ .

We study the average value of the divisor function $\unicode[STIX]{x1D70F}(n)$ for $n\leqslant x$ with $n\equiv a~\text{mod}~q$ . The divisor function is known to be evenly distributed over arithmetic progressions for all $q$ that are a little smaller than $x^{2/3}$ . We show how to go past this barrier when $q=p^{k}$ for odd primes $p$ and any fixed integer $k\geqslant 7$ .

Let $E(N)$ denote the number of positive integers $n\leqslant N$ , with $n\equiv 4\;(\text{mod}\;24)$ , which cannot be represented as the sum of four squares of primes. We establish that $E(N)\ll N^{11/32}$ , thus improving on an earlier result of Harman and the first author, where the exponent $7/20$ appears in place of $11/32$ .

In this paper we study the Duffin–Schaeffer conjecture, which claims that $\unicode[STIX]{x1D706}(\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }{\mathcal{E}}_{n})=1$ if and only if $\sum _{n=1}^{\infty }\unicode[STIX]{x1D706}({\mathcal{E}}_{n})=\infty$ , where $\unicode[STIX]{x1D706}$ denotes the Lebesgue measure on $\mathbb{R}/\mathbb{Z}$ ,

$$\begin{eqnarray}{\mathcal{E}}_{n}={\mathcal{E}}_{n}(\unicode[STIX]{x1D713})=\mathop{\bigcup }_{\substack{ m=1 \\ (m,n)=1}}^{n}\bigg(\frac{m-\unicode[STIX]{x1D713}(n)}{n},\frac{m+\unicode[STIX]{x1D713}(n)}{n}\bigg),\end{eqnarray}$$

$\unicode[STIX]{x1D713}$

$\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }{\mathcal{E}}_{n}$

$\bigcup _{n=1}^{\infty }{\mathcal{E}}_{n}$

$C>0$

$$\begin{eqnarray}\unicode[STIX]{x1D706}\bigg(\mathop{\bigcup }_{n=1}^{\infty }{\mathcal{E}}_{n}\bigg)\geqslant C\min \bigg\{\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D706}({\mathcal{E}}_{n}),1\bigg\}.\end{eqnarray}$$

$p$

In this paper, we study how small a box contains at least two points from a modular hyperbola $xy\equiv c\;(\text{mod}\;p)$ . There are two such points in a square of side length $p^{1/4+\unicode[STIX]{x1D716}}$ . Furthermore, it turns out that either there are two such points in a square of side length $p^{1/6+\unicode[STIX]{x1D716}}$ or the least quadratic non-residue is less than $p^{1/(6\sqrt{e})+\unicode[STIX]{x1D716}}$ .

An important result of Weyl states that for every sequence $(a_{n})_{n\geqslant 1}$ of distinct positive integers the sequence of fractional parts of $(a_{n}{\it\alpha})_{n\geqslant 1}$ is uniformly distributed modulo one for almost all ${\it\alpha}$ . However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy of $(\{a_{n}{\it\alpha}\})_{n\geqslant 1}$ for almost all ${\it\alpha}$ . In particular, it is very difficult to give sharp lower bounds for the speed of convergence. Until now this was only carried out for lacunary sequences $(a_{n})_{n\geqslant 1}$ and for some special cases such as the Kronecker sequence $(\{n{\it\alpha}\})_{n\geqslant 1}$ or the sequence $(\{n^{2}{\it\alpha}\})_{n\geqslant 1}$ . In the present paper we answer the question for a large class of sequences $(a_{n})_{n\geqslant 1}$ including as a special case all polynomials $a_{n}=P(n)$ with $P\in \mathbb{Z}[x]$ of degree at least 2.

The aim of the discrete logarithm problem with auxiliary inputs is to solve for ${\it\alpha}$ , given the elements $g,g^{{\it\alpha}},\ldots ,g^{{\it\alpha}^{d}}$ of a cyclic group $G=\langle g\rangle$ , of prime order  $p$ . The best-known algorithm, proposed by Cheon in 2006, solves for ${\it\alpha}$ in the case where $d\mid (p\pm 1)$ , with a running time of $O(\sqrt{p/d}+d^{i})$ group exponentiations ( $i=1$ or $1/2$ depending on the sign). There have been several attempts to generalize this algorithm to the case of ${\rm\Phi}_{k}(p)$ where $k\geqslant 3$ . However, it has been shown by Kim, Cheon and Lee that a better complexity cannot be achieved than that of the usual square root algorithms.

We propose a new algorithm for solving the DLPwAI. We show that this algorithm has a running time of $\widetilde{O}(\sqrt{p/{\it\tau}_{f}}+d)$ group exponentiations, where  ${\it\tau}_{f}$ is the number of absolutely irreducible factors of $f(x)-f(y)$ . We note that this number is always smaller than $\widetilde{O}(p^{1/2})$ .

In addition, we present an analysis of a non-uniform birthday problem.