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Let $K=\mathbb{Q}(i)$ be the field of Gaussian rationals. Let $\zeta_K(s)$ be the Dedekind zeta function associated to K, defined by $$\zeta_K(s):=\sum_{\mathfrak{a}\subseteq \mathbb{Z}[i]}\frac{1}{\mathrm{N}(\mathfrak{a})^s},$$ where $\mathfrak{a}$ ranges over all the nonzero ideals of $\mathbb{Z}[i]=\mathcal{O}_K$. It is a well-known fact that $\zeta_K(s)$ can be extended to a meromorphic function on $\mathbb{C}$ that has a simple pole at $s=1$ with residue $\pi/4$ and no other poles.

My question is that:

Can the following (finite) limit $$\lim_{s\to 1}\left(\zeta_K(s)-\dfrac{\pi}{4(s-1)}\right)$$ be explicitly computed (or at least in terms of the other constants, like the Euler-Mascheroni constant $\gamma$, etc)?

user26857
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Dat234
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  • You might be able to use the class number formula, along with the well-known Laurant series expansion for $\zeta(s)$ to get the whole Laurant series for $\zeta_K(s)$. Remember that since $\mathbb{Z}[i]$ is a PID that the class number is $1$. – user413766 Apr 16 '19 at 14:50
  • @JonHales The class number formula is in $\frac{\pi/4}{s-1}$ not in the constant term – reuns Apr 17 '19 at 10:13
  • For a general number field $K$, if you write $\zeta_K(s) = a/(s-1) + b + O(s-1)$ then the ratio $b/a$ has a name that you can look up: the Euler-Kronecker constant of $K$. Intrinsically, $b/a$ is the constant term in the Taylor expansion of the logarithmic derivative: $\zeta_K'(s)/\zeta_K(s) = -1/(s-1) + b/a + O(s-1)$. – KCd Apr 20 '19 at 06:05
  • If you have access to MathSciNet, you might look at the paper MR3378382 about Euler-Kronecker constants of quadratic fields. – KCd Apr 20 '19 at 06:12
  • @Dat234, if you are satisfied with the answer, please consider accepting it. – Arrow May 29 '19 at 17:39

1 Answers1

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From the factorization

$$\zeta_K(s)=\zeta(s)L(s,\chi)$$

where $\chi$ is the unique nontrivial character mod $4$, and the expansions

$$\zeta(s)=\frac1{s-1}+\gamma+\cdots$$

and

$$L(s,\chi)=L(1,\chi)+(s-1)L'(1,\chi)+\cdots$$

it follows that

$$\zeta_K(s)=\frac{L(1,\chi)}{s-1}+\gamma L(1,\chi)+L'(1,\chi)+\cdots$$

where $L(1,\chi)=\pi/4$ and

$$L'(1,\chi)=-\sum_{n=1}^\infty\frac{\chi(n)\log n}n.$$

This series can be computed explicitly using Kummer's formula for $\log \Gamma$.

user246336
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  • @reuns $L'(1,\chi)=\frac{1}{4} \pi \left(\gamma +\log \left(\frac{4 \pi ^3}{\Gamma \left(\frac{1}{4}\right)^4}\right)\right)$ – user246336 Apr 17 '19 at 10:29
  • So you meant the functional equation lets us express $L'(1,\chi)$ in term of $\gamma$ and $\sum_{a=1}^q \chi(a) \log \Gamma(a/q)$ – reuns Apr 17 '19 at 10:33