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In 1970 A. Baker and W. Schmidt introduced regular systems of numbers and vectors, showing that the set of real algebraic numbers forms a regular system on any fixed interval. This fact was used to prove several important results in the…

Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some $q$-series identity for proving the zeta function has an Euler product and then,…

We introduce the notion of saturated sets of primes of an algebraic number field and prove an analogue of Riemann's existence theorem for the decomposition groups of infinite stably saturated sets of primes.

The main result of this paper is an instance of the conjecture made by Gouvea and Mazur (Math. Res. Lett., 1995) which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k…

We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products…

Let $K/\mathbb{Q}$ be an algebraic extension of fields, and let $\alpha \not= 0$ be contained in an algebraic closure of $K$. If $\alpha$ can be approximated by roots of numbers in $K^{\times}$ with respect to the Weil height, we prove that…

We exploit transformations relating generalized $q$-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as $\pi$, and to connect sums…

Let $X$ be a variety over a finite field. Given an order $R$ in a semi-simple algebra over the rationals and a constructible \'etale sheaf $F$ of $R$-modules over $X$, one can consider a natural non-commutative $L$-function associated with…

In a recent paper [5] a smooth function f : [0; 1] --> R with all derivatives vanishing at 0 has been considered and a global condition, showing that f is indeed identically 0, has been presented. The purpose of this note is to replace the…

In this mostly expository note, we explain a proof of Tate's two conjectures [Tat65] for algebraic cycles of arbitrary codimension on certain products of elliptic curves and abelian surfaces over number fields.

As well known, the important hypothesis formulated by B.G. RIEMANN in 1859 states that all non-trivial zeroes of the Zeta function $Z(s)=\sum_{n=1}^{\infty } n^{-s}$ should fall on the Critical Line (C.L.) $Re(s)=\frac{1}{2}$.\\ Although…

We define a class of computable functions over real numbers using functional schemes similar to the class of primitive and partial recursive functions defined by G\"odel and Kleene. We show that this class of functions can also be…

Let k be a an algebraically closed field of arbitrary characteristic, and we let h be the usual Weil height for the n-dimensional affine space corresponding to the function field k(t) (extended to its algebraic closure). We prove that for…

There are many formulations of problems that have been proven to be equivalent to the Riemann Hypothesis in modern mathematics. In this paper we look at the formulation of an inequality derived by Robin in 1984 that proves the Riemann…

Given a number field $K \neq \mathbb{Q}$, in a now classic work, Stark pinpointed the possible source of a so-called Landau-Siegel zero of the Dedekind zeta function $\zeta_K(s)$ and used this to give effective upper and lower bounds on the…

We disprove the sum-product conjecture for real numbers by constructing arbitrarily large $A\subset \mathbb{R}$ (whose elements are algebraic integers in a number field of degree $\asymp \log\lvert A\rvert$) such that \[\max(\lvert…

In this paper after proving (in Section 2) the Berkovich analytic space analog of the familiar fact that there exist many non-isomorphic Riemann surfaces of the fixed topological type, I introduce the precise notion of Arithmetic…

Beginning from the resolution of the Dirichlet L function, using the inner product formula between two infinite-dimensional vectors in the complex space, the author proved the baffling problem--Hecke conjecture.

In his 1999 preprint "Universal Lie Algebra", P. Vogel put forward a hypothesis on the existence of a universal Lie algebra. Although this hypothesis remains open, it is known that many quantities in Lie theory admit universal descriptions.…

We identify the weights $wt(f_n)$ of a family $\{f_n\}$ of rotation symmetric Boolean functions with the cardinalities of the sets of $n$-periodic points of a finite-type shift, recovering the second author's result that said weights…