Extending arithmetic functions (and associated Dirichlet series) to arbitrary rings of integers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T17:34:31Z http://mathoverflow.net/feeds/question/108796 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108796/extending-arithmetic-functions-and-associated-dirichlet-series-to-arbitrary-rin Extending arithmetic functions (and associated Dirichlet series) to arbitrary rings of integers Steve Pandarus 2012-10-04T09:41:14Z 2012-10-04T11:24:23Z <p>Many classical arithmetic functions can be thought of as functions on the set of (non-zero) ideals of $\mathbb{Z}$ rather than as functions on $\mathbb{N}$.</p> <p>Example: For $n \in \mathbb{N}$ the divisor function $d(n)$ is defined to equal the number of divisors of $n$. Equivalently, we could define $d(I)$ to the number of ideals dividing the ideal $I \lhd \mathbb{Z}$, where we count once the fact that $(1)|(n)$.</p> <p>Now, the associated Dirichlet could be written as $$\sum^\infty_{n=1} d(n) n^{-s} = \sum_{I \lhd \mathbb{Z}} d(I) N(I)^{-s} = \zeta(s)^2.$$</p> <p>If $K$ is a number field with ring of integers $\mathcal{O}$ we can extend the definition of $d(I)$ to the ideals of $\mathcal{O}$. Furthermore, we can consider the associated Dirichlet series: $$\sum_{I \lhd \mathcal{O}} d(I) N(I)^{-s}.$$</p> <p>QUESTIONS: Is this Dirichlet series equal to $\zeta_K(s)^2$, the square of the Dedekind zeta function of the field? </p> <p>Does this phenomena generalize? ie if $a(n)$ is arithmetic function that can be equivalently defined on the ideals of $\mathbb{Z}$ and the Dirichlet series associated to $a(n)$ is a quotient of Riemann zeta functions (= Dedekind zeta function of $\mathbb{Q}$). Can we simply `replace' the Riemann zeta functions with the appropriate Dedekind zeta functions to obtain the Dirichet series associated to the extended a(n)? </p> http://mathoverflow.net/questions/108796/extending-arithmetic-functions-and-associated-dirichlet-series-to-arbitrary-rin/108806#108806 Answer by David Loeffler for Extending arithmetic functions (and associated Dirichlet series) to arbitrary rings of integers David Loeffler 2012-10-04T11:24:23Z 2012-10-04T11:24:23Z <p>The answer to your Question 1 is "yes". It's clear that the number of ideals of $\mathcal{O}$ of norm $\le M$ is bounded above by a polynomial in $M$, so one can manipulate Dirichlet series term-by-term for $Re(s) \gg 0$ and argue that</p> <p><code>$$\zeta_K(s)^2 = \sum_{A, B} N(A)^{-s} N(B)^{-s} = \sum_{C} \#\{ (A, B) : AB = C\} N(C)^{-s} = \sum_C d(C) N(C)^{-s}.$$</code></p> <p>As for Question 2 it's not entirely clear to me what your precise question is, but philosophically at least the answer is "yes" -- any arithmetical function $a$ definable purely in terms of ideals of $\mathbb{Z}$ will have a natural generalization to ideals of a number field, and if you can express $\sum_n a(n) n^{-s}$ in terms of the Riemann zeta then the same argument should give you an expression for $\sum_{I \triangleleft \mathcal{O}} a(I) N(I)^{-s}$ in terms of $\zeta_K(s)$. </p>