A homothetic arithmetic function of ratio $K$ is a function $f:\mathbb{N}\to R$ such that $f\left(Kn\right)=f\left(n\right)$ for every $n\in \mathbb{N}$. Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of $f\left(\mathbb{N}\right)$ in terms of the period and the ratio of $f$.

Let $a$ and $b\in \mathbb{N}$. Denote by $R_{a,b}$ the set of all integers $n>1$ whose canonical prime representation $n=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_r^{{\alpha }_r}$ has all exponents ${\alpha }_i$ $(1\le i\le r)$ being a multiple of $a$ or belonging to the arithmetic progression $at+b$, $t\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$. All integers in $R_{a,b}$ are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given. ...

In this paper, we give a numerical characterization of nef arithmetic $ℝ$-Cartier divisors of $C^0$-type on an arithmetic surface. Namely an arithmetic $ℝ$-Cartier divisor $\overline{D}$ of $C^0$-type is nef if and only if $\overline{D}$ is pseudo-effective and $\widehat{\mathrm{deg}}\left({\overline{D}}^2\right)=\widehat{\mathrm{vol}}\left(\overline{D}\right)$.

Let $(𝒪,\Sigma ,F_{\infty })$ be an arithmetic ring of Krull dimension at most 1, $𝒮=\mathrm{Spec}𝒪$ and $(\pi :𝒳\to 𝒮;{\sigma }_1,...,{\sigma }_n)$ an $n$-pointed stable curve of genus $g$. Write $𝒰=𝒳\setminus {\cup }_j{\sigma }_j\left(𝒮\right)$. The invertible sheaf ${\omega }_{𝒳/𝒮}({\sigma }_1+\cdots +{\sigma }_n)$ inherits a hermitian structure ${\parallel ·\parallel }_{\mathrm{hyp}}$ from the dual of the hyperbolic metric on the Riemann surface ${𝒰}_{\infty }$. In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ${\omega }_{𝒳/𝒮}{({\sigma }_1+...+{\sigma }_n)}_{\mathrm{hyp}}$. The theorem is applied to modular curves $X\left(\Gamma \right)$, $\Gamma ={\Gamma }_0\left(p\right)$ or ${\Gamma }_1\left(p\right)$, $p\ge 11$ prime, with sections given by the cusps. We show $Z^{\text{'}}(Y\left(\Gamma \right),1)\sim e^a{\pi }^b{\Gamma }_2{(1/2)}^{c}L(0,{ℳ}_{\Gamma })$, with $p\equiv 11mod12$ when $\Gamma ={\Gamma }_0\left(p\right)$. Here $Z\left(Y\right(\Gamma ),s)$ is the Selberg...

A geometric progression of length k and integer ratio is a set of numbers of the form $a,ar,...,a{r}^{k-1}$ for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence ${\left(a_i\right)}_{i=1}^{\infty }$ of positive real numbers with a₁ = 1 such that the set $G^{\left(k\right)}={\bigcup }_{i=1}^{\infty }(a_{2i},a_{2i-1}]$ contains no geometric progression of length k and integer ratio. Moreover, $G^{\left(k\right)}$ is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that...

Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of $X_0^{+}\left(p^r\right)\left(\mathbb{Q}\right)$, for $r\>1$ and $p$ a prime number exceeding $2·{10}^{11}$. This includes the case of the curves $X_{\mathrm{split}}\left(p\right)$. We then prove, with the help of computer calculations, that the same holds true for $p$ in the range $11\le p\le {10}^{14}$, $p\ne 13$. The combination of those results completes the qualitative study of rational points on $X_0^{+}\left(p^r\right)$ undertook in our previous work, with the only exception of $p^r={13}^2$.

We prove that every set A ⊂ ℤ satisfying ${\sum }_{x}min(1_A*1_A\left(x\right),t)\le (2+\delta )t\left|A\right|$ for t and δ in suitable ranges must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset A ⊂ ℕ satisfies |ℕ∖(A+A)| ≥ k; specifically, we show that $\mathbb{P}\left(\right|\mathbb{N}\setminus (A+A)\left|\ge k\right)=\Theta \left(2^{-k/2}\right)$.

We construct an explicit family of arithmetic Teichmüller curves ${𝒞}_{2k}$, $k\in \mathbb{N}$, supporting $\mathrm{SL}(2,ℝ)$-invariant probabilities ${\mu }_{2k}$ such that the associated $\mathrm{SL}(2,ℝ)$-representation on $L^2({𝒞}_{2k},{\mu }_{2k})$ has complementary series for every $k\ge 3$. Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichmüller geodesic flow restricted to these explicit arithmetic Teichmüller curves ${𝒞}_{2k}$ has arbitrarily slow rate of exponential mixing.

We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over ℚ and p be a prime of good reduction for E. Let $e_E\left(p\right)$ be the exponent of the group of rational points of the reduction modulo p of E over the finite field ${}_p$. Let be the family of elliptic curves $E_{a,b}:y^2=x^3+ax+b$, where |a| ≤ A and |b| ≤ B. We prove that, for any c > 1 and k∈ ℕ, $1/\left|\right|{\sum }_{E\in }{\sum }_{p\le x}{e}_E^k\left(p\right)=C_{k}li\left(x^{k+1}\right)+O(\left(x^{k+1}\right)/{\left(logx\right)}^c$)$$as x → ∞, as long...

Let K be a number field, X be a smooth projective curve over it and D be a reduced divisor on X. Let (E,∇) be a vector bundle with connection having meromorphic singularities on D. Let $p_1,...,p_s\in X\left(K\right)$ and $X^o:=X̅\setminus D,p_1,...,p_s$ (the $p_j$’s may be in the support of D). Using tools from Nevanlinna theory and formal geometry, we give the definition of E-section of arithmetic type of the vector bundle E with respect to the points $p_j$; this is the natural generalization of the notion of E-function defined in Siegel-Shidlovskiĭ...