Abel's summation formula

{{wikibooks|Analytic Number Theory/Useful summation formulas}}

Let $(a\_n)\_\{n=0\}^\backslash infty$ be a sequence of real or complex numbers. Define the partial sum function $A$ by

:$A(t)\; =\; \backslash sum\_\{0\; \backslash le\; n\; \backslash le\; t\}\; a\_n$

for any real number $t$. Fix real numbers , and let $\backslash phi$ be a continuously differentiable function on $[x,\; y]$. Then:

:

The formula is derived by applying integration by parts for a Riemann–Stieltjes integral to the functions $A$ and $\backslash phi$.

Taking the left endpoint to be $-1$ gives the formula

:$\backslash sum\_\{0\; \backslash le\; n\; \backslash le\; x\}\; a\_n\backslash phi(n)\; =\; A(x)\backslash phi(x)\; -\; \backslash int\_0^x\; A(u)\backslash phi\text{'}(u)\backslash ,du.$

If the sequence $(a\_n)$ is indexed starting at $n\; =\; 1$, then we may formally define $a\_0\; =\; 0$. The previous formula becomes

:$\backslash sum\_\{1\; \backslash le\; n\; \backslash le\; x\}\; a\_n\backslash phi(n)\; =\; A(x)\backslash phi(x)\; -\; \backslash int\_1^x\; A(u)\backslash phi\text{'}(u)\backslash ,du.$

A common way to apply Abel's summation formula is to take the limit of one of these formulas as $x\; \backslash to\; \backslash infty$. The resulting formulas are

:$\backslash begin\{align\}$

\sum_{n=0}^\infty a_n\phi(n) &= \lim_{x \to \infty}\bigl(A(x)\phi(x)\bigr) - \int_0^\infty A(u)\phi'(u)\,du, \\

\sum_{n=1}^\infty a_n\phi(n) &= \lim_{x \to \infty}\bigl(A(x)\phi(x)\bigr) - \int_1^\infty A(u)\phi'(u)\,du.

\end{align}

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

A particularly useful case is the sequence $a\_n\; =\; 1$ for all $n\; \backslash ge\; 0$. In this case, $A(x)\; =\; \backslash lfloor\; x\; +\; 1\; \backslash rfloor$. For this sequence, Abel's summation formula simplifies to

:$\backslash sum\_\{0\; \backslash le\; n\; \backslash le\; x\}\; \backslash phi(n)\; =\; \backslash lfloor\; x\; +\; 1\; \backslash rfloor\backslash phi(x)\; -\; \backslash int\_0^x\; \backslash lfloor\; u\; +\; 1\backslash rfloor\; \backslash phi\text{'}(u)\backslash ,du.$

Similarly, for the sequence $a\_0\; =\; 0$ and $a\_n\; =\; 1$ for all $n\; \backslash ge\; 1$, the formula becomes

:$\backslash sum\_\{1\; \backslash le\; n\; \backslash le\; x\}\; \backslash phi(n)\; =\; \backslash lfloor\; x\; \backslash rfloor\backslash phi(x)\; -\; \backslash int\_1^x\; \backslash lfloor\; u\; \backslash rfloor\; \backslash phi\text{'}(u)\backslash ,du.$

Upon taking the limit as $x\; \backslash to\; \backslash infty$, we find

:$\backslash begin\{align\}$

\sum_{n=0}^\infty \phi(n) &= \lim_{x \to \infty}\bigl(\lfloor x + 1 \rfloor\phi(x)\bigr) - \int_0^\infty \lfloor u + 1\rfloor \phi'(u)\,du, \\

\sum_{n=1}^\infty \phi(n) &= \lim_{x \to \infty}\bigl(\lfloor x \rfloor\phi(x)\bigr) - \int_1^\infty \lfloor u\rfloor \phi'(u)\,du,

\end{align}

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

Abel's summation formula can be generalized to the case where $\backslash phi$ is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral:

:

By taking $\backslash phi$ to be the partial sum function associated to some sequence, this leads to the summation by parts formula.

If $a\_n\; =\; 1$ for $n\; \backslash ge\; 1$ and $\backslash phi(x)\; =\; 1/x,$ then $A(x)\; =\; \backslash lfloor\; x\; \backslash rfloor$ and the formula yields

:$\backslash sum\_\{n=1\}^\{\backslash lfloor\; x\; \backslash rfloor\}\; \backslash frac\{1\}\{n\}\; =\; \backslash frac\{\backslash lfloor\; x\; \backslash rfloor\}\{x\}\; +\; \backslash int\_1^x\; \backslash frac\{\backslash lfloor\; u\; \backslash rfloor\}\{u^2\}\; \backslash ,du.$

The left-hand side is the harmonic number $H\_\{\backslash lfloor\; x\; \backslash rfloor\}$.

Fix a complex number $s$. If $a\_n\; =\; 1$ for $n\; \backslash ge\; 1$ and $\backslash phi(x)\; =\; x^\{-s\},$ then $A(x)\; =\; \backslash lfloor\; x\; \backslash rfloor$ and the formula becomes

:$\backslash sum\_\{n=1\}^\{\backslash lfloor\; x\; \backslash rfloor\}\; \backslash frac\{1\}\{n^s\}\; =\; \backslash frac\{\backslash lfloor\; x\; \backslash rfloor\}\{x^s\}\; +\; s\backslash int\_1^x\; \backslash frac\{\backslash lfloor\; u\backslash rfloor\}\{u^\{1+s\}\}\backslash ,du.$

If $\backslash Re(s)\; >\; 1$, then the limit as $x\; \backslash to\; \backslash infty$ exists and yields the formula

:$\backslash zeta(s)\; =\; s\backslash int\_1^\backslash infty\; \backslash frac\{\backslash lfloor\; u\backslash rfloor\}\{u^\{1+s\}\}\backslash ,du.$

where $\backslash zeta(s)$ is the Riemann zeta function.

This may be used to derive Dirichlet's theorem that $\backslash zeta(s)$ has a simple pole with residue 1 at {{math|s {{=}} 1}}.

The technique of the previous example may also be applied to other Dirichlet series. If $a\_n\; =\; \backslash mu(n)$ is the Möbius function and $\backslash phi(x)\; =\; x^\{-s\}$, then $A(x)\; =\; M(x)\; =\; \backslash sum\_\{n\; \backslash le\; x\}\; \backslash mu(n)$ is Mertens function and

:$\backslash frac\{1\}\{\backslash zeta(s)\}\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash mu(n)\}\{n^s\}\; =\; s\backslash int\_1^\backslash infty\; \backslash frac\{M(u)\}\{u^\{1+s\}\}\backslash ,du.$

This formula holds for $\backslash Re(s)\; >\; 1$.

• Summation by parts
• Integration by parts

• {{citation|first=Tom|last=Apostol|authorlink=Tom Apostol|title=Introduction to Analytic Number Theory|publisher=Springer-Verlag|series=Undergraduate Texts in Mathematics|year=1976}}.

Category:Number theory

Category:Summability methods