Abel's summation formula

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 $\phi (x)$ a function of class ${\mathcal {C}}^{1}$ . Then

\sum _{1\leq n\leq x}a_{n}\phi (n)=A(x)\phi (x)-\int _{1}^{x}A(u)\phi '(u)\,\mathrm {d} u

where

A(x):= \sum_{1 \le n \le x} a_n \,.

More generally, we have

\sum_{x< n\le y} a_n\phi(n) = A(y)\phi(y) - A(x)\phi(x) -\int_x^y A(u)\phi'(u)\,\mathrm{d}u \,.

If $a_{n}=1$ and $\phi (x) = \frac{1}{x} \,,$ then $A(x)=\lfloor x\rfloor$ and

\sum_1^x \frac{1}{n} = \frac{\lfloor x \rfloor}{x} + \int_1^x \frac{\lfloor u \rfloor}{u^2} \, \mathrm{d}u

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

If $a_{n}=1$ and $\phi (x) = \frac{1}{x^s} \,,$ then $A(x)=\lfloor x\rfloor$ and

\sum_1^\infty \frac{1}{n^s} = s\int_1^\infty \frac{\lfloor u\rfloor}{u^{1+s}} \mathrm{d}u \,.

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

If $a_{n}=\mu (n)$ is the Möbius function and $\phi (x) = \frac{1}{x^s} \,,$ then $A(x)=M(x)=\sum _{n\leq x}\mu (n)$ is Mertens function and

\sum_1^\infty \frac{\mu(n)}{n^s} = s \int_1^\infty \frac{M(u)}{u^{1+s}} \mathrm{d}u \,.

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

Apostol, Tom (1976), Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag .

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