Analytic Number Theory/Dirichlet series

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For the remainder of this book, we shall use Riemann's convention of denoting complex numbers:

Definition[edit | edit source]

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[edit | edit source]

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 , .


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 .

Theorem 5.3 (abscissa of conditional convergence):

Formulas[edit | edit source]

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



This follows directly from theorem 2.11 and the fact that strongly multiplicative strongly multiplicative.