Abel's summation formula - Alchetron, the free social encyclopedia

In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in number theory to compute series.

Identity

Let a n be a sequence of real or complex numbers and ϕ ( x ) a function of class C 1 . Then

∑ 1 ≤ n ≤ x a n ϕ ( n ) = A ( x ) ϕ ( x ) − ∫ 1 x A ( u ) ϕ ′ ( u ) d u

where

A ( x ) := ∑ 1 ≤ n ≤ x a n .

Indeed, this is integration by parts for a Riemann–Stieltjes integral.

More generally, we have

∑ x < n ≤ y a n ϕ ( n ) = A ( y ) ϕ ( y ) − A ( x ) ϕ ( x ) − ∫ x y A ( u ) ϕ ′ ( u ) d u .

If a n = 1 and ϕ ( x ) = 1 x , then A ( x ) = ⌊ x ⌋ and

∑ 1 x 1 n = ⌊ x ⌋ x + ∫ 1 x ⌊ u ⌋ u 2 d u

which is a method to represent the Euler–Mascheroni constant.

If a n = 1 and ϕ ( x ) = 1 x s , then A ( x ) = ⌊ x ⌋ and

∑ 1 ∞ 1 n s = s ∫ 1 ∞ ⌊ u ⌋ u 1 + s d u .

The formula holds for ℜ ( s ) > 1 . It may be used to derive Dirichlet's theorem, that is, ζ ( s ) has a simple pole with residue 1 in s = 1.

If a n = μ ( n ) is the Möbius function and ϕ ( x ) = 1 x s , then A ( x ) = M ( x ) = ∑ n ≤ x μ ( n ) is Mertens function and

∑ 1 ∞ μ ( n ) n s = s ∫ 1 ∞ M ( u ) u 1 + s d u .

This formula holds for ℜ ( s ) > 1 .

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