TY - JOUR

T1 - On the sum of consecutive integers in sequences

AU - Tsai, Mu Tsun

AU - Zaharescu, Alexandru

N1 - Funding Information:
The research of the second author was supported in part by the National Science Foundation Grant DMS-0901621.

PY - 2012/5

Y1 - 2012/5

N2 - 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 ℕ(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(n), (4) (assuming the Riemann Hypothesis) f(n) = ||n|| := min{|n - p| : p ∈ ℙ} (the distance to the set of primes).

AB - 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 ℕ(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(n), (4) (assuming the Riemann Hypothesis) f(n) = ||n|| := min{|n - p| : p ∈ ℙ} (the distance to the set of primes).

KW - Partition

KW - Riemann Hypothesis

KW - consecutive integers

KW - prime gap

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U2 - 10.1142/S1793042112500364

DO - 10.1142/S1793042112500364

M3 - Article

AN - SCOPUS:84859054286

SN - 1793-0421

VL - 8

SP - 643

EP - 652

JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 3

ER -