This paper deals with applications of Voronin’s universality theorem for the Riemann zeta-function $\zeta$. Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values $\zeta(\sigma+it)$ for real t where $\sigma\in(1/2,1)$ is fixed. In this sense, the values of the zeta-function on any such vertical line provides an atlas for plane curves. In the same framework, we study the curvature of curves generated from $\zeta(\sigma+it)$ when $\sigma>1/2$ and we show that there is a connection with the zeros of $\zeta'(\sigma+it)$. Moreover, we clarify under which conditions the real and the imaginary part of the zeta-function are jointly universal.

Erdős, Graham and Selfridge considered, for each positive integer n, the least value of $t_n$ so that the integers $n+1, n+2, \dots, n+t_n $ contain a subset the product of whose members with n is a square. An open problem posed by Granville concerns the size of $t_n$, under the assumption of the ABC conjecture. We establish some results on the distribution of $t_n$, and in the process solve Granville’s problem unconditionally.

In this paper, together with the preceding Part I [10], we develop a framework for tame geometry on Henselian valued fields of characteristic zero, called Hensel minimality. It adds to [10] the treatment of the mixed characteristic case. Hensel minimality is inspired by o-minimality and its role in real geometry and diophantine applications. We develop geometric results and applications for Hensel minimal structures that were previously known only under stronger or less axiomatic assumptions, and which often have counterparts in o-minimal structures. We prove a Jacobian property, a strong form of Taylor approximations of definable functions, resplendency results and cell decomposition, all under Hensel minimality – more precisely, $1$-h-minimality. We obtain a diophantine application of counting rational points of bounded height on Hensel minimal curves.

A crucial ingredient in the theory of theta liftings of Kudla and Millson is the construction of a $q$-form $\varphi_{KM}$ on an orthogonal symmetric space, using Howe's differential operators. This form can be seen as a Thom form of a real oriented vector bundle. We show that the Kudla-Millson form can be recovered from a canonical construction of Mathai and Quillen. A similar result was obtaind by Garcia for signature $(2,q)$ in case the symmetric space is hermitian and we extend it to arbitrary signature.

This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_{p}$-extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$, where $\Gamma$ is the Galois group of the $\mathbb{Z}_{p}$-extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_{p}$-extension, then it holds for almost all $\mathbb{Z}_{p}$-extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p-adic Lie extension.

For any subset $Z \subseteq {\mathbb {Q}}$, consider the set $S_Z$ of subfields $L\subseteq {\overline {\mathbb {Q}}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in L such that $C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$, we show that if Z is not thin in ${\mathbb {Q}}$, then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here, thin and meager both mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers $\mathcal {O}_L$ is universally definable in L. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists $-definable subset of an algebraic extension of ${\mathbb Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$. As a consequence of Katz–Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q.

In this article, we prove that the Riemann hypothesis implies a conjecture of Chandee on shifted moments of the Riemann zeta function. The proof is based on ideas of Harper concerning sharp upper bounds for the $2k$th moments of the Riemann zeta function on the critical line.

We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty )$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.

We prove discrete restriction estimates for a broad class of hypersurfaces arising in seminal work of Birch. To do so, we use a variant of Bourgain’s arithmetic version of the Tomas–Stein method and Magyar’s decomposition of the Fourier transform of the indicator function of the integer points on a hypersurface.

Let $S=\{s_{1}, s_{2}, \ldots \}$ be an unbounded sequence of positive integers with $s_{n+1}/s_{n}$ approaching $\alpha $ as $n\rightarrow \infty $ and let $\beta>\max (\alpha , 2)$. We show that for all sufficiently large positive integers l, if $A\subset [0, l]$ with $l\in A$, $\gcd A=1$ and $|A|\geq (2-{k}/{\lambda \beta })l/(\lambda +1)$, where $\lambda =\lceil {k}/{\beta }\rceil $, then $kA\cap S\neq \emptyset $ for $2<\beta \leq 3$ and $k\geq {2\beta }/{(\beta -2)}$ or for $\beta>3$ and $k\geq 3$.

We study the étale cohomology of Hilbert modular varieties, building on the methods introduced by Caraiani and Scholze for unitary Shimura varieties. We obtain the analogous vanishing theorem: in the ‘generic’ case, the cohomology with torsion coefficients is concentrated in the middle degree. We also probe the structure of the cohomology beyond the generic case, obtaining bounds on the range of degrees where cohomology with torsion coefficients can be non-zero. The proof is based on the geometric Jacquet–Langlands functoriality established by Tian and Xiao and avoids trace formula computations for the cohomology of Igusa varieties. As an application, we show that, when $p$ splits completely in the totally real field and under certain technical assumptions, the $p$-adic local Langlands correspondence for $\mathrm {GL}_2(\mathbb {Q}_p)$ occurs in the completed homology of Hilbert modular varieties.

We derive an explicit formula for the N-point correlation $F_N(s)$ of the van der Corput sequence in base $2$ for all $N \in \mathbb {N}$ and $s \geq 0$. The formula can be evaluated without explicit knowledge about the elements of the van der Corput sequence. This constitutes the first example of an exact closed-form expression of $F_N(s)$ for all $N \in \mathbb {N}$ and all $s \geq 0$ which does not require explicit knowledge about the involved sequence. Moreover, it can be immediately read off that $\lim _{N \to \infty } F_N(s)$ exists only for $0 \leq s \leq 1/2$.

In this paper, we present the results related to a problem posed by Andrzej Schinzel. Does the number $N_1(n)$ of integer solutions of the equation

$$ \begin{align*}x_1+x_2+\cdots+x_n=x_1x_2\cdot\ldots\cdot x_n,\,\,x_1\ge x_2\ge\cdots\ge x_n\ge 1\end{align*} $$

$N_a(n)$

$$ \begin{align*}x_1+x_2+\cdots+x_n=ax_1x_2\cdot\ldots\cdot x_n,\,\, x_1\ge x_2\ge\cdots\ge x_n\ge 1.\end{align*} $$

$N_2(n)=1$

$2n-3$

$N_2(n)$

$\{n:N_2(n)\le k,\,n\ge 2\}$

In this article, we give generalizations of the well-known Fermat’s Little Theorem, Wilson’s theorem, and the little-known Gegenbauer’s theorem.

Let $\ell $ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell $-towers of multigraphs. In this context, growth patterns are realized by certain analogs of Iwasawa invariants, which depend on the prime $\ell $ and the abelian $\ell $-tower of multigraphs. We formulate and study statistical questions about the behavior of the Iwasawa $\mu $ and $\lambda $ invariants.

Let G be a semiabelian variety defined over an algebraically closed field K of prime characteristic. We describe the intersection of a subvariety X of G with a finitely generated subgroup of $G(K)$.

In our paper, we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq {\mathbb {Z}}/q{\mathbb {Z}}$ in the case of composite q. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.

Given a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $\gamma$ of elements of S, one can associate an arboreal representation to $\gamma$, generalising the case of iterating a single polynomial. We study the probability that a random sequence $\gamma$ produces a “large-image” representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets S defined over $\mathbb{Z}[t]$, and we conjecture a similar positive-probability result for suitable sets over $\mathbb{Q}$. As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all S possessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration.

Let $k \geqslant 2$ be an integer. We prove that factorisation of integers into k parts follows the Dirichlet distribution $\mathrm{Dir}\left({1}/{k},\ldots,{1}/{k}\right)$ by multidimensional contour integration, thereby generalising the Deshouillers–Dress–Tenenbaum (DDT) arcsine law on divisors where $k=2$. The same holds for factorisation of polynomials or permutations. Dirichlet distribution with arbitrary parameters can be modelled similarly.