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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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