Quasimodular forms generalize modular forms and have been studied actively in recent years in connection with various topics in number theory and geometry. One of their interesting properties is that they correspond to finite sequences of modular forms of certain types. We extend such a correspondence to the case of Hilbert quasimodular forms. As an application we construct Poincaré series for Hilbert quasimodular forms.

In this article we introduce Besov-type spaces with variable smoothness and integrability, which unify and generalize the Besov-type spaces with fixed exponents. Under natural regularity assumptions on the exponent functions, we show that our spaces are well-defined, i.e., independent of the choice of basis functions and we establish some properties of these function spaces. Moreover the Sobolev embeddings for these function spaces are obtained.

We say that a set $\mathcal{S}$ is additively decomposed into two sets $\mathcal{A}$ and $\mathcal{B}$ if $\mathcal{S} = \{a+b: a\in \mathcal{A}, b \in \mathcal{B}\}$. A. Sárközy has recently conjectured that the set $\mathcal{Q}$ of quadratic residues modulo a prime $p$ does not have nontrivial decompositions. Although various partial results towards this conjecture have been obtained, it is still open. Here we obtain a nontrivial upper bound on the number of such decompositions.

We compute the Fourier expansion of Hecke operators on vector-valued modular forms for the Weil representation associated to a lattice $L$. The Hecke operators considered in this paper include operators $T(p^{2l})$ where $p$ is a prime dividing the level of the lattice $L$. Additionally, an explicit formula for a general type of Gauss sum associated to a lattice $L$ drops out as a by-product.

The purpose of this article is consider $|\zeta'(\sigma + it)/\zeta(\sigma + it)|$ and $|\zeta(\sigma + it)|^{-1}$ when $\sigma$ is close to unity. We prove that $|\zeta'(\sigma + it)/\zeta(\sigma + it)| \leq 87\log t$ and $|\zeta(\sigma + it)|^{-1} \leq 6.9\times 10^{6} \log t$ for $\sigma \geq 1-1/(8 \log t)$ and $t\geq 45$.

Letting $\Omega(n)$ denote the number of prime factors of $n$ counted with multiplicity, Rivat, Sárközy and Stewart (1999) proved a result regarding maximal cardinalities of sets ${\cal A},{\cal B}\subset\{1,\ldots,N\}$ so that for every $a\in{\cal A}$ and $b\in{\cal B}$, $\Omega(a+b)$ is even. This paper extends their work in several directions. The role of $\lambda(n)=(-1)^{\Omega(n)}$ is generalized to all non-constant completely multiplicative functions $f:\mathbb{N}\rightarrow \{-1,1\}$. Rather than just $\Omega$ being even on ${\cal A}+{\cal B}$, we extend the result to all possible parities of $\Omega$ on ${\cal A}$, ${\cal B}$, and ${\cal A}+{\cal B}$. Furthermore, we prove that many such pairs $({\cal A},{\cal B})$ exist. Results from Ramsey theory and extremal graph theory are used.

Let $p \ge 5$ be a prime and $E$ an elliptic curve without complex multiplication and let $K_\infty=\mathbb{Q}(E[p^\infty])$ be a pro-$p$ Galois extension over a number field $K$. We consider $X(E/\K_\infty)$, the Pontryagin dual of the $p$-Selmer group $Sel_{p^\infty}(E/K_\infty)$. The size of this module is roughly measured by its rank $\tau$ over a $p$-adic Galois group algebra $\Lambda(H)$, which has been studied in the past decade. We prove $\tau \ge 2$ for almost every elliptic curve under standard assumptions. We find that $\tau = 1$ and $j \notin \mathbb{Z}$ is impossible, while $\tau = 1$ and $j \in \mathbb{Z}$ can occur in at most $8$ explicitly known elliptic curves. The rarity of $\tau=1$ was expected from Iwasawa theory, but the proof is essentially elementary. It follows from a result of Coates et al. that $\tau$ is odd if and only if $[\mathbb{Q}(E[p]) : \mathbb{Q}]/2$ is odd. We show that this is equivalent to $p=7$, $E$ having a $7$-isogeny, a simple condition on the discriminant and local conditions at $2$ and $3$. Up to isogeny, these curves are parametrised by two rational variables using recent work of Greenberg, Rubin, Silverberg and Stoll.

Let $p$ be a prime number of the form $p=2\ell^f+1$ with an odd prime number $\ell$, and $h_p^-$ the relative class number of the $p$th cyclotomic field $K=\mathbb{Q}(\zeta_p)$. When $f=1$, it is conjectured that $h_p^-$ is odd, and there are several results related to this conjecture. In this paper, we deal with the case $f \geq 2$. For $0 \leq t \leq f$, let $h_{p,t}^-$ denote the relative class number of the imaginary subfield $K_t$ of $K$ of degree $2\ell^t$ over $\mathbb{Q}$. We show that the ratio $h_{p,f}^-/h_{p,f-1}^-$ is not divisible by a prime number $r$ if $r$ is a primitive root modulo $\ell^2$. Further, when $r \leq 47$, we give some computational results on the ratio $h_{p,t}^-/h_{p,t-1}^-$ for $1 \leq t \leq f$. In the range of our computation, we find that the ratio is divisible by $r$ only in some exceptional cases.

We use duality in convex analysis and particularly the famous Attouch-Brezis theorem to prove subdifferentiability of infimal convolution on Banach couples.

Given an additive function $f$ and a multiplicative function $g$, let $E(f,g;x)=\#\{n\le x: f(n)=g(n)\}$. We study the size of $E(f,g;x)$ for functions $f$ such that $f(n)\neq 0$ for at least one integer $n>1$. In particular, we show that for those additive functions $f$ whose values $f(n)$ are concentrated around their mean value $\lambda(n)$, one can find a multiplicative function $g$ such that, given any $\varepsilon>0$, then $E(f,g;x)\gg x/\lambda(x)^{1+\varepsilon}$. We also show that given any additive function satisfying certain regularity conditions, no multiplicative function can coincide with it on a set of positive density. It follows that if $\omega(n)$ stands for the number of distinct prime factors of $n$, then, given any $\varepsilon>0$, there exists a multiplicative function $g$ such that $E(\omega,g;x)\gg x/(\log\log x)^{1+\varepsilon}$, while for all multiplicative functions $g$, we have $E(\omega,g;x)=o(x)$ as $x\to \infty$.