Search

Given positive integers n, n′ and k, we investigate the Möbius-Bernoulli numbers Mk(n), double Möbius-Bernoulli numbers Mk(n,n′), and Möbius-Bernoulli polynomials Mk(n)(x). We find new identities involving double Möbius-Bernoulli, Barnes-Bernoulli numbers and Dedekind sums. In part of this paper, the Möbius-Bernoulli polynomials Mk(n)(x), can be interpreted as critical values of the following Dirichlet type L-function $$\begin{array}{} \displaystyle L_{HM}(s;n,x):=\sum_{d|n} \sum_{m= 0}^\infty \frac{\mu(d)}{(md+x)^s} \, \, \text{(for Re} (s) \gt 1), \end{array} $$ which has analytic continuation to the whole s-complex plane, where μ is the Möbius function.

We prove that the reciprocal sum $S_k(x)$ of the least common multiple of $k\geq 3$ positive integers in $\N\cap [1,x]$ satisfies $$ S_k(x)=P_{2^k-1}(\log x)+O(x^{-\theta_k+\epsilon}) $$ where $P$ is a polynomial of degree $2^k-1$ and $\theta_k=\frac{2^k}{(k+1)^{\frac{k+1}2}}\cdot \frac3{2^k+6k-5}$. This was conjectured in Hilberdink, Luca, and T\'{o}th~\cite[Remark 2.4]{HLT}. We also prove asymptotic formulas for similar sums conjectured there., Comment: 16 pages, 2 figures, an error in the previous version in section 3.3 is fixed, the results are made weaker

In 1949, Motzkin proved that every Euclidean domain $R$ has a minimal Euclidean function, $\phi_R$. He showed that when $R = \mathbb{Z}$, the minimal function is $\phi_{\mathbb{Z}}(x) = \lfloor \log_2 |x| \rfloor$. For over seventy years, $\phi_{\mathbb{Z}}$ has been the only example of an explictly-computed minimal function in a number field. We give the first explicitly-computed minimal function in a non-trivial number field, $\phi_{\mathbb{Z}[i]}$, which computes the length of the shortest possible $(1+i)$-ary expansion of any Gaussian integer. We also present an algorithm that uses $\phi_{\mathbb{Z}[i]}$ to compute minimal $(1+i)$-ary expansions of Gaussian integers. We solve these problems using only elementary methods., Comment: 10 pages, 5 figures

Lenstra introduced the notion of Euclidean ideal classes for number fields to study cyclicity of their class groups. In particular, he showed that the class group of a number field with unit rank greater than or equal to one is cyclic if and only if it has a Euclidean ideal class. The only if part in the above result is conditional on the extended Riemann hypothesis. Graves and Murty showed that one does not require the extended Riemann hypothesis if the unit rank of the number field is greater than or equal to four and its Hilbert class field is abelian over rationals. In this article, we study real cubic and quadratic fields with cyclic class groups and show that they have a Euclidean ideal class under certain conditions.

This article gives a closed-form expression for the determinant of binary circulant matrices.

We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representing prime numbers by the binary quadratic form x2 + 3xy + y2 . The Euclidean algorithm is commenced with inputs from one of the families, and the first remainder less than a predetermined size produces the modular inverse or representation.

We introduce a beautiful order of the rational quaternion algebra(-1,-3)/ℚ , which has an interesting structure like the Hurwitz order. We show that the order is norm euclidean by a geometrical method, which is quite simple. We also show that the quadratic form x²+y²+3z²+3w² is universal, using this order, in an elementary way.

Let $(u_n)_{n \geq 0}$ be a non-degenerate Lucas sequence, given by the relation $u_n=a_1 u_{n-1}+a_2 u_{n-2}$. Let $\ell _u(m)={\rm lcm}(m, z_u(m))$, for $(m,a_2)=1$, where $z_u(m)$ is the rank of appearance of $m$ in $u_n$. We prove that $$ \sum _{\substack {m>x\\ (m,a_2)=1}}\frac {1}{\ell _u(m)}\leq \exp \biggl (\!-\biggl (\frac 1{\sqrt {6}}-\varepsilon +o(1)\biggr )\sqrt {(\log x)(\log \log x)}\biggr ), $$ when $x$ is sufficiently large in terms of $\varepsilon $, and where the $o(1)$ depends on $u$. Moreover, if $g_u(n)=\gcd (n,u_n)$, we show that for every $k\geq 1$, $$ \sum _{n\leq x}g_u(n)^{k}\leq x^{k+1}\exp (-(1+o(1)) \sqrt {(\log x)(\log \log x)}), $$ when $x$ is sufficiently large, and where the $o(1)$ depends upon $u$ and $k$. This gives a partial answer to a question posed by C. Sanna. As a by-product, we derive bounds on $\#\{n\leq x: (n, u_n)>y\}$, at least in certain ranges of $y$, which strengthens what was already obtained by Sanna. Finally, we begin the study of the multiplicative analogs of $\ell _u(m)$, finding interesting results.

Let f be an arithmetic function and S = {x 1, …, xn } be a set of n distinct positive integers. By (f(xi , xj )) (resp. (f[xi , xj ])) we denote the n × n matrix having f evaluated at the greatest common divisor (xi , xj ) (resp. the least common multiple [xi , xj ]) of x, and xj as its (i, j)-entry, respectively. The set S is said to be gcd closed if (xi , xj ) ∈ S for 1 ≤ i, j ≤ n. In this paper, we give formulas for the determinants of the matrices (f(xi , xj )) and (f[xi , xj ]) if S consists of multiple coprime gcd-closed sets (i.e., S equals the union of S 1, …, Sk with k ≥ 1 being an integer and S 1, …, Sk being gcd-closed sets such that (lcm(Si ), lcm(Sj )) = 1 for all 1 ≤ i ≠ j ≤ k). This extends the Bourque-Ligh, Hong’s and the Hong-Loewy formulas obtained in 1993, 2002 and 2011, respectively. It also generalizes the famous Smith’s determinant.

We study integral solutions $n$ of the equation $A(n+k)=A(n)$, where $A$ is a certain multiplicative function related to Jordan's totient function.

In this paper, we study the derivatives of an integer-valued polynomial of a given degree. Denoting by $E_n$ the set of the integer-valued polynomials with degree $\leq n$, we show that the smallest positive integer $c_n$ satisfying the property: $\forall P \in E_n, c_n P' \in E_n$ is $c_n = \mathrm{lcm}(1 , 2 , \dots , n)$. As an application, we deduce an easy proof of the well-known inequality $\mathrm{lcm}(1 , 2 , \dots , n) \geq 2^{n - 1}$ ($\forall n \geq 1$). In the second part of the paper, we generalize our result for the derivative of a given order $k$ and then we give two divisibility properties for the obtained numbers $c_{n , k}$ (generalizing the $c_n$'s). Leaning on this study, we conclude the paper by determining, for a given natural number $n$, the smallest positive integer $\lambda_n$ satisfying the property: $\forall P \in E_n$, $\forall k \in \mathbb{N}$: $\lambda_n P^{(k)} \in E_n$. In particular, we show that: $\lambda_n = \prod_{p \text{ prime}} p^{\lfloor\frac{n}{p}\rfloor}$ ($\forall n \in \mathbb{N}$)., Comment: 17 pages

Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. We prove that $\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x$ for all $x \geq 2$, and that $\mathcal{A}$ has zero asymptotic density. Our proofs rely on a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$.

This paper introduces two classes of totally real quartic number fields, one of biquadratic extensions and one of cyclic extensions, each of which has a non-principal Euclidean ideal. It generalizes techniques of Graves used to prove that the number field $\mathbb{Q}(\sqrt{2},\sqrt{35})$ has a non-principal Euclidean ideal., 12 pages

In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.

In this paper we introduce a connected topology T on the set ℕ of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are connected in the topology T form a basis for Golomb’s topology. Further we examine connectedness of arithmetic progressions in the division topology T′ on ℕ which was defined by Rizza in 1993. Immediate consequences of these studies are results concerning local connectedness of the topological spaces (ℕ, T) and (ℕ, T′).

We shall deal with infinite towers of cyclic fields of genus number 1 in which a prime number $l\geq 5$ is totally ramified. Our main result states that, if $m$ is a positive divisor of $l-1$ less than $(l-1)/2$, then for any positive integer $n$, the cyclic field of degree $ml^{n}$ with conductor $l^{n+1}$ is not norm-Euclidean. In particular, it follows that, for any positive integer $n$, the (real) cyclic field of degree $l^{n}$ with conductor $l^{n+1}$ is not norm-Euclidean and that the (imaginary) cyclic field of degree 14 with conductor 49, whose class number is known to equal 1, is not norm-Euclidean.

The paper deals with asymptotics for a class of arithmetic functions which describe the value distribution of the greatest-common-divisor function. Typically, they are generated by a Dirichlet series whose analytic behavior is determined by the factor ζ2(s)ζ(2s − 1). Furthermore, multivariate generalizations are considered.

For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 = 2 be any integer. In this paper, we show that, for integers alpha,r >= a and n >= 2*alpha*r, we have L_n >= u_0*r^{alpha+a-2}*(r+1)^n. In particular, letting a = 2 yields an improvement to the best previous lower bound on L_n (obtained by Hong and Yang) for all but three choices of alpha,r >= 2., 5 pages; v3: improved results and updated references

The mosaic of the integer [math] is the array of prime numbers resulting from iterating the Fundamental Theorem of Arithmetic on [math] and on any resulting composite exponents. In this paper, we generalize several number theoretic functions to the mosaic of [math] , first based on the primes of the mosaic, second by examining several possible definitions of a divisor in terms of mosaics. Having done so, we examine properties of these functions.

For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm $x$ or less as $x$ goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with explicit error terms. We also present results for higher moments along with computational data which support the results for the second, third, fourth, and fifth moments. The analogous question for integers is studied by Diaconis and Erd\"os., Comment: 13 pages, 4 figures

It is known that certain combinatorial convolution sums involving two divisor functions product formulae of arbitrary level can be explicitly expressed as a linear combination of divisor functions. In this article we deal with cases for certain combinatorial convolution sums involving three, four, six and twelve divisor functions product formula and obtain explicit expressions.

The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series which involves a product of Riemann zeta-functions of a special form., Comment: 11 pages

For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathrm{lcm}(u_0,u_1,\ldots, u_n)$ of the finite arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower bounds on $L_n$ which improve upon those obtained previously when either $u_0$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n+1$ for $u_0$ properly chosen, and is also nearly sharp as $n\to\infty$., Comment: 10 pages

Reverse analyses of three proofs of the Sylvester-Gallai theorem lead to three different and incompatible axiom systems. In particular, we show that proofs respecting the purity of the method, using only notions considered to be part of the statement of the theorem to be proved, are not always the simplest, as they may require axioms which proofs using extraneous predicates do not rely upon.

In this paper, we prove the identity $$\lcm\{\binom{k}{0}, \binom{k}{1}, >..., \binom{k}{k}\} = \frac{\lcm(1, 2, ..., k, k + 1)}{k + 1} (\forall k \in \mathbb{N}) .$$ As an application, we give an easily proof of the well-known nontrivial lower bound $\lcm(1, 2, ..., k) \geq 2^{k - 1}$ $(\forall k \geq 1)$., 5 pages. To appear in American Mathematical Monthly