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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

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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

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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 .

Theorem 5.3 (abscissa of conditional convergence):

Formulas

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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.