Abel’s summation formula – Wikipedia

Integration by parts version of Abel’s method for summation by parts

In mathematics, Abel’s summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series.

Formula[edit]

Let

${displaystyle (a_{n})_{n=0}^{infty }}$

be a sequence of real or complex numbers. Define the partial sum function

${displaystyle A}$

by

${displaystyle A(t)=sum _{0leq nleq t}a_{n}}$

for any real number

${displaystyle t}$

. Fix real numbers

${displaystyle x$

, and let

${displaystyle phi }$

be a continuously differentiable function on

${displaystyle [x,y]}$

. Then:

${displaystyle sum _{x$

The formula is derived by applying integration by parts for a Riemann–Stieltjes integral to the functions

${displaystyle A}$

and

${displaystyle phi }$

.

Variations[edit]

Taking the left endpoint to be

${displaystyle -1}$

gives the formula

${displaystyle sum _{0leq nleq x}a_{n}phi (n)=A(x)phi (x)-int _{0}^{x}A(u)phi ‘(u),du.}$

If the sequence

${displaystyle (a_{n})}$

is indexed starting at

${displaystyle n=1}$

, then we may formally define

${displaystyle a_{0}=0}$

. The previous formula becomes

${displaystyle sum _{1leq nleq x}a_{n}phi (n)=A(x)phi (x)-int _{1}^{x}A(u)phi ‘(u),du.}$

A common way to apply Abel’s summation formula is to take the limit of one of these formulas as

${displaystyle xto infty }$

. The resulting formulas are

${displaystyle {begin{aligned}sum _{n=0}^{infty }a_{n}phi (n)&=lim _{xto infty }{bigl (}A(x)phi (x){bigr )}-int _{0}^{infty }A(u)phi ‘(u),du,\sum _{n=1}^{infty }a_{n}phi (n)&=lim _{xto infty }{bigl (}A(x)phi (x){bigr )}-int _{1}^{infty }A(u)phi ‘(u),du.end{aligned}}}$

These equations hold whenever both limits on the right-hand side exist and are finite.

A particularly useful case is the sequence

${displaystyle a_{n}=1}$

for all

${displaystyle ngeq 0}$

. In this case,

${displaystyle A(x)=lfloor x+1rfloor }$

. For this sequence, Abel’s summation formula simplifies to

${displaystyle sum _{0leq nleq x}phi (n)=lfloor x+1rfloor phi (x)-int _{0}^{x}lfloor u+1rfloor phi ‘(u),du.}$

Similarly, for the sequence

${displaystyle a_{0}=0}$

and

${displaystyle a_{n}=1}$

for all

${displaystyle ngeq 1}$

, the formula becomes

${displaystyle sum _{1leq nleq x}phi (n)=lfloor xrfloor phi (x)-int _{1}^{x}lfloor urfloor phi ‘(u),du.}$

Upon taking the limit as

${displaystyle xto infty }$

, we find

${displaystyle {begin{aligned}sum _{n=0}^{infty }phi (n)&=lim _{xto infty }{bigl (}lfloor x+1rfloor phi (x){bigr )}-int _{0}^{infty }lfloor u+1rfloor phi ‘(u),du,\sum _{n=1}^{infty }phi (n)&=lim _{xto infty }{bigl (}lfloor xrfloor phi (x){bigr )}-int _{1}^{infty }lfloor urfloor phi ‘(u),du,end{aligned}}}$

assuming that both terms on the right-hand side exist and are finite.

Abel’s summation formula can be generalized to the case where

${displaystyle phi }$

is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral:

${displaystyle sum _{x$

By taking

${displaystyle phi }$

to be the partial sum function associated to some sequence, this leads to the summation by parts formula.

Examples[edit]

Harmonic numbers[edit]

If

${displaystyle a_{n}=1}$

for

${displaystyle ngeq 1}$

and

${displaystyle phi (x)=1/x,}$

then

${displaystyle A(x)=lfloor xrfloor }$

and the formula yields

${displaystyle sum _{n=1}^{lfloor xrfloor }{frac {1}{n}}={frac {lfloor xrfloor }{x}}+int _{1}^{x}{frac {lfloor urfloor }{u^{2}}},du.}$

The left-hand side is the harmonic number

${displaystyle H_{lfloor xrfloor }}$

.

Representation of Riemann’s zeta function[edit]

Fix a complex number

${displaystyle s}$

. If

${displaystyle a_{n}=1}$

for

${displaystyle ngeq 1}$

and

${displaystyle phi (x)=x^{-s},}$

then

${displaystyle A(x)=lfloor xrfloor }$

and the formula becomes

${displaystyle sum _{n=1}^{lfloor xrfloor }{frac {1}{n^{s}}}={frac {lfloor xrfloor }{x^{s}}}+sint _{1}^{x}{frac {lfloor urfloor }{u^{1+s}}},du.}$

If

${displaystyle Re (s)>1}$

${displaystyle xto infty }$

exists and yields the formula

${displaystyle zeta (s)=sint _{1}^{infty }{frac {lfloor urfloor }{u^{1+s}}},du.}$

This may be used to derive Dirichlet’s theorem that

${displaystyle zeta (s)}$

has a simple pole with residue 1 at s = 1.

Reciprocal of Riemann zeta function[edit]

The technique of the previous example may also be applied to other Dirichlet series. If

${displaystyle a_{n}=mu (n)}$

is the Möbius function and

${displaystyle phi (x)=x^{-s}}$

, then

${displaystyle A(x)=M(x)=sum _{nleq x}mu (n)}$

is Mertens function and

${displaystyle {frac {1}{zeta (s)}}=sum _{n=1}^{infty }{frac {mu (n)}{n^{s}}}=sint _{1}^{infty }{frac {M(u)}{u^{1+s}}},du.}$

This formula holds for

${displaystyle Re (s)>1}$

See also[edit]

References[edit]