The Mordell–Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell–Lang...

Let $D_m$ be an elliptic curve over $\mathbb{Q}$ of the form $y^2=x^3-m^{2}x+m^2$, where $m$ is an integer. In this paper we prove that the two points $P_{-1}=(-m,m)$ and $P_0=(0,m)$ on $D_m$ can be extended to a basis for $D_m\left(\mathbb{Q}\right)$ under certain conditions described explicitly.

It is shown that the multiplicative independence of Dedekind zeta functions of abelian fields is equivalent to their functional independence. We also give all the possible multiplicative dependence relations for any set of Dedekind zeta functions of abelian fields.

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{|}^p$1/p$,$aj ∈ ℂ$,$such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ and L > 0 let ${\kappa }_{\infty }(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0|\ge Lma{x}_{1\le j\le n}|a_j|$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that $c_1\surd (n/L)-1\le {\kappa }_{\infty }(n,L)\le c_2\surd (n/L)$ for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈ (0,1]. Essentially...

In this paper we study the structure and the degeneracies of the Mumford-Tate group $MT\left(M\right)$ of a 1-motive $M$ defined over $ℂ$. This group is an algebraic $\mathbb{Q}$- group acting on the Hodge realization of $M$ and endowed with an increasing filtration $W_{•}$. We prove that the unipotent radical of $MT\left(M\right)$, which is $W_{-1}\left(MT\left(M\right)\right)$, injects into a “generalized” Heisenberg group. We then explain how to reduce to the study of the Mumford-Tate group of a direct sum of 1-motives whose torus’character group and whose lattice are both of rank 1....

A new derivation of the classic asymptotic expansion of the $n$-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994).Realistic bounds for the error with ${li}^{-1}\left(n\right)$, after having retained the first $m$ terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n\ge r_3$, we have $p_n\>s_3\left(n\right)$ where $s_3\left(n\right)$ is the sum of the first four terms of the asymptotic expansion.

The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of $t$-cores, and give an elementary proof by using the Macdonald identities. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function...

It is shown, that the function$$\begin{array}{cc}\hfill H\left(x\right)& =\sum _{k=-\infty }^{\infty }{e}^{-k^{2}x}\hfill \\ \multicolumn{2}{c}{\text{satisfies}\text{the}\text{relation}}\\ \hfill H\left(x\right)& =\sum _{n=0}^{\infty }\frac{{\left(2\pi \right)}^{2n}}{\left(2n\right)!}{H}^{\left(n\right)}\left(x\right).\hfill \end{array}$$

Previous work by Rubinstein and Gao computed the n-level densities for families of quadratic Dirichlet L-functions for test functions f̂₁, ..., f̂ₙ supported in ${\sum }_{i=1}^n\left|u_i\right|<2$, and showed agreement with random matrix theory predictions in this range for n ≤ 3 but only in a restricted range for larger n. We extend these results and show agreement for n ≤ 7, and reduce higher n to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form,...