2 6 Ja n 20 04 Common divisors of a n − 1 and b n − 1 over function fields

ar X iv : m at h / 04 01 12 6 v 1 [ m at h . N T ] 1 3 Ja n 20 04 THREE LECTURES ON THE RIEMANN ZETA - FUNCTION

On the Number of Points of Some Kummer Curves over Finite Fields

Let l be a prime number and let k = Fq be a finite field of characteristic p 6= l with q = p elements. Let n ≥ 1. The curve Cn : y l = x(x n − 1) is smooth of genus gn = φ(l) 2 = l(l−1) 2 over k. Let Fn/k be the function field of Cn, let PFn denote the set of places, and let DivFn denote the group of divisors of Fn/k. The absolute norm N(P) of a place P ∈ PFn is the cardinality of its residue c...

A remark on the means of the number of divisors

‎We obtain the asymptotic expansion of the sequence with general term $frac{A_n}{G_n}$‎, ‎where $A_n$ and $G_n$ are the arithmetic and geometric means of the numbers $d(1),d(2),dots,d(n)$‎, ‎with $d(n)$ denoting the number of positive divisors of $n$‎. ‎Also‎, ‎we obtain some explicit bounds concerning $G_n$ and $frac{A_n}{G_n}$.

Compositions versus Cyclic Compositions

We prove that the sum of greatest common divisors of parts in all compositions of n equals the sum of lengths of all cyclic compositions of n. The proof highlights structural similarities between the set of compositions of n and the set of cyclic compositions of n. 1. Definitions and Statement A composition of the positive integer n is an ordered partition of n; that is, a vector ( ) r p p ...,...

A Submodule-Based Zero Divisors Graph for Modules

‎Let $R$ be commutative ring with identity and $M$ be an $R$-module‎. ‎The zero divisor graph of $M$ is denoted $Gamma{(M)}$‎. ‎In this study‎, ‎we are going to generalize the zero divisor graph $Gamma(M)$ to submodule-based zero divisor graph $Gamma(M‎, ‎N)$ by replacing elements whose product is zero with elements whose product is in some submodules $N$ of $M$‎. ‎The main objective of this pa...