On the sum of consecutive integers in sequences

Mu Tsun Tsai, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review

Abstract

Let A be a set of natural numbers, r A(n) be the number of ways to represent n as a sum of consecutive elements in A, and M A(x) := ∑ n ≤ x r A(n). Under various circumstances, we show that M (x) ∼ ∑ M Ai(x), where ℕ = square cup sign -A i is a partition of ℕ. In particular, we prove that the asymptotic formula holds when the A i's are chosen such that A i = {n ∈ ℕ: f(n) = i}, where f(n) can be one of the following: (1) f(n) = Ω(n), (2) f(n) = d(n), (3) f(n) = r (n), (4) (assuming the Riemann Hypothesis) f(n) = ||n|| := min{|n - p| : p ∈ ℙ} (the distance to the set of primes).

Original languageEnglish (US)
Pages (from-to)643-652
Number of pages10
JournalInternational Journal of Number Theory
Volume8
Issue number3
DOIs
StatePublished - May 1 2012

Keywords

  • Partition
  • Riemann Hypothesis
  • consecutive integers
  • prime gap

ASJC Scopus subject areas

  • Algebra and Number Theory

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