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Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions

Siao Hong, Shuangnian Hu, Shaofang Hong (2016)

Open Mathematics

Let f be an arithmetic function and S = {x1, …, 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...

Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

Kevin J. McGown (2012)

Journal de Théorie des Nombres de Bordeaux

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant Δ = 7 2 , 9 2 , 13 2 , 19 2 , 31 2 , 37 2 , 43 2 , 61 2 , 67 2 , 103 2 , 109 2 , 127 2 , 157 2 . A large part of the proof is in establishing the following more general result: Let K be a Galois number field of odd prime degree ℓ and conductor f . Assume the GRH for ζ K ( s ) . If 38 ( ℓ - 1 ) 2 ( log f ) 6 log log f < f , then K is not norm-Euclidean.

Note on some greatest common divisor matrices

Peter Lindqvist, Kristian Seip (1998)

Acta Arithmetica

Some quadratic forms related to "greatest common divisor matrices" are represented in terms of L²-norms of rather simple functions. Our formula is especially useful when the size of the matrix grows, and we will study the asymptotic behaviour of the smallest and largest eigenvalues. Indeed, a sharp bound in terms of the zeta function is obtained. Our leading example is a hybrid between Hilbert's matrix and Smith's matrix.

On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

Manfred Kühleitner, Werner Nowak (2013)

Open Mathematics

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 of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

Given an integer n ≥ 3 , let u 1 , ... , u n be pairwise coprime integers ≥ 2 , 𝒟 a family of nonempty proper subsets of { 1 , ... , n } with “enough” elements, and ε a function 𝒟 → { ± 1 } . Does there exist at least one prime q such that q divides ∏ i ∈ I u i - ε ( I ) for some I ∈ 𝒟 , but it does not divide u 1 ⋯ u n ? We answer this question in the positive when the u i are prime powers and ε and 𝒟 are subjected to certain restrictions.We use the result to prove that, if ε 0 ∈ { ± 1 } and A is a set of three or more primes that contains all prime divisors of any number of the form ∏ p ∈ B p - ε 0 for...

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