Analytic Number Theory/Dirichlet series

For the remainder of this book, we shall use Riemann's convention of denoting complex numbers: 0 100% #FFFFFF Definition 0 100% #FFFFFF Definition 5.1: Let be an arithmetic function. Then the Dirichlet series associated to is the series , where ranges over the complex numbers. Convergence considerations 0 100% #FFFFFF Theorem 5.2 (abscissa of absolute convergence): Let be an arithmetic function such that the series of absolute values associated to the Dirichlet series associated to neither diverges at all nor converges for all . Then there exists , called the abscissa of absolute convergence, such that the Dirichlet series associated to converges absolutely for all , and it's associated series of absolute values diverges for all , . Proof: Denote by the set of all real numbers such that diverges. Due to the assumption, this set is neither empty nor equal to . Further, if , then for all and all , since and due to the comparison test. It follows that has a supremum. Let be that supremum. By definition, for we have convergence, and if we had convergence for we would have found a lower upper bound due to the above argument, contradicting the definition of . 0 100% #FFFFFF Theorem 5.3 (abscissa of conditional convergence): Formulas 0 100% #FFFFFF Theorem 8.4 (Euler product): Let be a strongly multiplicative function, and let such that the corresponding Dirichlet series converges absolutely. Then for that series we have the formula . Proof: This follows directly from theorem 2.11 and the fact that strongly multiplicative strongly multiplicative.