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Ternary quadratic forms with rational zeros

John Friedlander, Henryk Iwaniec (2010)

Journal de Théorie des Nombres de Bordeaux

We consider the Legendre quadratic forms ϕ a b ( x , y , z ) = a x 2 + b y 2 - z 2 and, in particular, a question posed by J–P. Serre, to count the number of pairs of integers 1 ≤ a ≤ A , 1 ≤ b ≤ B , for which the form ϕ a b has a non-trivial rational zero. Under certain mild conditions on the integers a , b , we are able to find the asymptotic formula for the number of such forms.

The asymptotic behaviour of the counting functions of Ω-sets in arithmetical semigroups

Maciej Radziejewski (2014)

Acta Arithmetica

We consider an axiomatically-defined class of arithmetical semigroups that we call simple L-semigroups. This class includes all generalized Hilbert semigroups, in particular the semigroup of non-zero integers in any algebraic number field. We show, for all positive integers k, that the counting function of the set of elements with at most k distinct factorization lengths in such a semigroup has oscillations of logarithmic frequency and size √ x ( l o g x ) - M for some M>0. More generally, we show a result on...

The distribution of square-free numbers of the form [ n c ]

Xiaodong Cao, Wenguang Zhai (1998)

Journal de théorie des nombres de Bordeaux

It is proved that the sequence [ n c ] ( n = 1 , 2 , ⋯ ) contains infinite squarefree integers whenever 1 < c < 61 36 = 1 . 6944 ⋯ , which improves Rieger’s earlier range 1 < c < 1 . 5 .

Topological aspects of infinitude of primes in arithmetic progressions

František Marko, Štefan Porubský (2015)

Colloquium Mathematicae

We investigate properties of coset topologies on commutative domains with an identity, in particular, the 𝓢-coprime topologies defined by Marko and Porubský (2012) and akin to the topology defined by Furstenberg (1955) in his proof of the infinitude of rational primes. We extend results about the infinitude of prime or maximal ideals related to the Dirichlet theorem on the infinitude of primes from Knopfmacher and Porubský (1997), and correct some results from that paper. Then we determine cluster...

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