Abel's summation formula

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. Let be a sequence of real or complex numbers. Define the partial sum function by for any real number . Fix real numbers , and let be a continuously differentiable function on . Then: The formula is derived by applying integration by parts for a Riemann–Stieltjes integral to the functions and . Taking the left endpoint to be gives the formula If the sequence is indexed starting at , then we may formally define . The previous formula becomes A common way to apply Abel's summation formula is to take the limit of one of these formulas as . The resulting formulas are These equations hold whenever both limits on the right-hand side exist and are finite. A particularly useful case is the sequence for all . In this case, . For this sequence, Abel's summation formula simplifies to Similarly, for the sequence and for all , the formula becomes Upon taking the limit as , we find assuming that both terms on the right-hand side exist and are finite. Abel's summation formula can be generalized to the case where is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral: By taking to be the partial sum function associated to some sequence, this leads to the summation by parts formula. If for and then and the formula yields The left-hand side is the harmonic number . Fix a complex number . If for and then and the formula becomes If , then the limit as exists and yields the formula where is the Riemann zeta function. This may be used to derive Dirichlet's theorem that has a simple pole with residue 1 at s = 1. The technique of the previous example may also be applied to other Dirichlet series. If is the Möbius function and , then is Mertens function and This formula holds for .