zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Further improvements of lower bounds for the least common multiples of arithmetic progressions. (English) Zbl 1196.11007
For relatively prime positive integers $u_0$ and $r$, the authors consider the arithmetic progression $ \{u_k:=u_0+kr\}_{k=0}^n$. Define $L_n:=\text{lcm}\{u_0, u_1, \dots , u_n\}$ and let $ a\ge 2$ be any integer. In this paper they show that for integers $ \alpha, r\geq a$ and $ n\geq 2\alpha r$ $$ L_n\geq u_0r^{\alpha +a-2}(r+1)^n.$$ In particular, letting $ a=2$ yields an improvement to the best previous lower bound on $ L_n$ (obtained by {\it S. Hong} and {\it Y. Yang} [Proc. Am. Math. Soc. 136, No. 12, 4111--4114 (2008; Zbl 1157.11001)]) for all but three choices of $ \alpha , r\geq 2$.

11A05Multiplicative structure of the integers
11B25Arithmetic progressions
Full Text: DOI arXiv