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On the Euler-Kronecker constants of global fields and primes with small norms. (English) Zbl 1185.11069
Ginzburg, Victor (ed.), Algebraic geometry and number theory. In Honor of Vladimir Drinfeld’s 50th birthday. Basel: Birkhäuser (ISBN 978-0-8176-4471-0/hbk). Progress in Mathematics 253, 407-451 (2006).
Let \(K\) be a global field, i.e., either an algebraic number field of finite degree, or an algebraic function field of one variable over a finite field. Let \(\zeta_K(s)\) be the Dedekind zeta function of \(K\), with the Laurent expansion at \(s=1\): \[ \zeta_K(s)=c_{-1}(s-1)^{-1}+c_0+c_1(s-1)+\ldots . \] The author presents a systematic study of the real number \[ \gamma_K=\frac{c_0}{c_{-1}}, \] which he calls the Euler-Kronecker constant of \(K\). Let \[ \Phi_K(x)=\frac{1}{x-1}\sum_{N(P)^k\leq x}\left(\frac{x}{N(P)^k}-1\right)\log N(P)\quad (x>1), \] where \(P\) is a (non-archimedean) prime divisor of \(K\) with the norm \(N(P)\) and \(k\geq 1\).
Suppose that \(K\) is a number field of degree \(n=[K:\mathbb Q]\) and discriminant \(d_K\). If the Generalized Riemann Hypothesis (GRH) is true then \[ \gamma_K<\left(\frac{\alpha_K+1}{\alpha_K-1}\right)(2\log\alpha_K+1-\Phi_K(\alpha_K^2))\leq\left(\frac{\alpha_K+1}{\alpha_K-1}\right)(2\log\alpha_K+1), \] provided that \(n>2\), or \(n=2\) and \(|d_K|>8\). Here \[ \alpha_K=\frac12 \log|d_K|. \] Let \[ \beta_K=-\frac{r_1}{2}(\gamma+\log 4\pi)-r_2(\gamma+\log 2\pi), \] with \(r_1, r_2\) the number of real and imaginary places of \(K\), \(\gamma\) Euler’s constant. Then, unconditionally, \[ \gamma_K>-\alpha_K-\beta_K-1. \] Let \(n>1\) and \[ \alpha_K^*=\frac{\alpha_K}{n-1}. \] If \(\alpha_K^*>1\) then, under GRH, \[ \frac{\alpha_K^*+1}{\alpha_K^*-1}(\gamma_K+1)>-2(n-1)(\log\alpha_K^*+1). \]
Suppose that \(K\) is a function field of one variable with the constant field \({\mathbb F}_q\) and genus \(g\). Then \[ \gamma_K<\left(\frac{\alpha_K+1}{\alpha_K-1}\right)(2\log\alpha_K+1+\log q-\Phi_K(\alpha_K^2))\leq \]
\[ \leq\left(\frac{\alpha_K+1}{\alpha_K-1}\right)(2\log\alpha_K+1+\log q). \] Here \[ \alpha_K=(g-1)\log q. \] It holds that \[ \gamma_K>-\alpha_K-\frac{q+1}{2(q-1)}\log q. \] For fixed \(q\): \[ \lim\inf\frac{\gamma_K}{(g_K-1)\log q}\geq -\frac{1}{\sqrt{q}+1}. \] If \(K\) is an extension of \({\mathbb F}_q(t)\) of degree \(n>1\) and \[ \alpha_K^*:=\frac{(g-1)\log q}{n-1}>1 \] then \[ \frac{\alpha_K^*+1}{\alpha_K^*-1}(\gamma_K+\frac{q+1}{2(q-1)}\log q)>-2(n-1)(\log \alpha_K^*+\frac{\alpha_K^*}{\alpha_K^*-1}). \]
For the entire collection see [Zbl 1113.00007].

MSC:
11R42 Zeta functions and \(L\)-functions of number fields
11R47 Other analytic theory
11R58 Arithmetic theory of algebraic function fields
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