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The Riemann zeta function is defined as: $$ \zeta(s) := \sum_{n=1}^{\infty}\frac{1}{n^s} $$ for all $s$ in the $\textrm{Re}(s)>1$ half-plane.

In order to distinguish between $\sum_{n=1}^{\infty}\frac{1}{n^s}$ and the Euler product expression of $\zeta$, I usually refer to $\sum_{n=1}^{\infty}\frac{1}{n^s}$ as the Dirichlet series expression of $\zeta$.

The same applies for Dirichlet $L$-series: $$ L(\chi,s) = \sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}. $$ $\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}$ is the Dirichlet series expression of the Dirichlet $L$-series $L(\chi,s)$ (again, as distinct from the Euler product expression).

Now we come the the Dedekind zeta function, defined in terms of some algebraic number field $K$: $$ \zeta_K(s) := \sum_{\mathfrak{a}}\frac{1}{N(\mathfrak{a})^s}, $$ where the sum is taken over all integral ideals $\mathfrak{a} \subset \mathcal{O}_K$, and $N$ denotes the ideal norm in $\mathcal{O}_K$.

My question is: What do we call $\sum_{\mathfrak{a}}\frac{1}{N(\mathfrak{a})^s}$? Note that it is not a Dirichlet series.

Might we call it the additive series expression (again, as distinct from the Euler product expression)?

Any suggestions, or does anyone know of a universally accepted standard name for it?

Many thanks.

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    $\begingroup$ It can be viewed as a Dirichlet series if you group $N(\cdot)=n$ for all $n\in\mathbb Z^+$. $\endgroup$ Commented Apr 7, 2020 at 6:59
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    $\begingroup$ I don't see there being much difficulty in calling it a "Dirichlet series" (particularly if the terms are regrouped as suggested by Kemono Chen), though if you are concerned about this, perhaps "Dirichlet-type series" would do? For comparison, the term "Dirichlet-type integral" is used in the text Fractal Zeta Functions and Relative Fractal Drums by Lapidus et al. to refer to integrals of the type $\int \varphi(x)^s\,\mathrm{d}\mu(x)$. $\endgroup$ Commented Apr 7, 2020 at 15:27

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Call it a "Dirichlet series indexed by the ideals of $K$" (really, by the "nonzero ideals of $\mathcal O_K$").

In a similar way, the product representation $$ \zeta_K(s) = \prod_{\mathfrak p} \frac{1}{1 - 1/{\rm N}(\mathfrak p)^s} $$ for ${\rm Re}(s) > 1$ is an "Euler product indexed by the primes of $K$" (really, by the "nonzero prime ideals of $\mathcal O_K$"). This should be distinguished from an Euler product over rational primes, which in this case would be $$ \zeta_K(s) = \prod_{p} \frac{1}{Q_p(1/p^s)} $$ for $Q_p(1/p^s) = \prod_{\mathfrak p\mid p} (1 - 1/{\rm N}(\mathfrak p)^s)$. Over $\mathbf Q$, a function represented by a Dirichlet series or Euler product on some right half-plane has a unique such expression, but over a number field bigger than $\mathbf Q$ there are many different ideals (including different prime ideals) with the same norm, so a Dirichlet series or Euler product over a number field bigger than $\mathbf Q$ can arise in more than one way (it does not determine its coefficients).

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