A formula for the Dedekind \(\xi\)-function of an imaginary quadratic field (Q5945099)

Let \(k\) be an imaginary quadratic field of discriminant \(d\). The Dedekind zeta function \(\zeta_k(s)\) of \(k\) can be written as \(\zeta_k(s)=\zeta(s)L(s, \chi_d)\), where \(L(s, \chi_d)\) is the Dirichlet \(L\)-function of character \(\chi_d=({d\over\cdot})\). The Dedekind \(\xi\)-function of \(k\) is defined by NEWLINE\[NEWLINE\xi_k(s)=s(s-1)|d|^{s\over 2}(2\pi)^{-s}\Gamma(s)\zeta(s)L(s, \chi_d).NEWLINE\]NEWLINE Let us recall that two quadratic forms \(F(x, y)=ax^2+bxy+cy^2\) and \(F'(x, y)=a'x^2+b'xy+c'y^2\) are equivalent if an element \(\gamma\in SL_2(Z)\) exists such that NEWLINE\[NEWLINE\begin{pmatrix} a'&b'/2\\ b'/2&c'\end{pmatrix}= \gamma^t\begin{pmatrix} a&b/2\\ b/2&c\end{pmatrix}\gamma,NEWLINE\]NEWLINE where \(\gamma^t\) is the transpose of \(\gamma\). Let \(w\) be the number of roots of unity of \(k\). Let NEWLINE\[NEWLINE\Phi(t)=\sum_{\{F\}}\sum_{m,n=-\infty}^\infty{\pi F(m, n)\over\sqrt{|d|}} \left({\pi t F(m, n)\over\sqrt{|d|}}-1\right)\exp \left(-{2\pi t F(m, n)\over\sqrt{|d|}}\right),NEWLINE\]NEWLINE where the first sum is over the inequivalent classes of positive definite integral quadratic forms of discriminant \(d\). In the paper under review, the author obtains that NEWLINE\[NEWLINE\xi_k(s)={4\over w}\int_1^\infty\Phi(t)(t^s+t^{1-s}) dt \tag \(*\) NEWLINE\]NEWLINE for all complex \(s\). This generalizes the well-known formula for the Riemann \(\xi\)-function: NEWLINE\[NEWLINE\xi(s)=4\int_1^\infty\left[x^{3\over 2}\psi'(x)\right]'(x^{s-1\over 2}+x^{-{ s\over 2}})dx, \quad\text{where}\quad \psi(x)=\sum_{n=1}^{\infty}e^{-\pi n^2 x}.NEWLINE\]NEWLINE From (\(*\)) he derives that NEWLINE\[NEWLINEw\xi_k\left(1\over 2\right)\geq\sum_{a, b, c}\left({\sqrt{|d|}\over 2a}\right)^{1/2} \left[1-\gamma(a)-{1\over 2}\ln{\sqrt{|d|}\over 2a}\right]- {1\over 2}\sum_{a, b, c}{\sqrt{|d|}\over 2c}e^{-{\pi c\over\sqrt{|d|}}},NEWLINE\]NEWLINE where the sum is over quadratic forms \((a, b, c)\) of discriminant \(d=b^2-4ac\) with \(-a<b\leq a<c\) or \(0\leq b\leq a=c\) and NEWLINE\[NEWLINE\gamma(a)=\int_1^\infty\psi(t){dt\over\sqrt{t}}+ \int_1^{{\sqrt{|d|}\over 2a}}\psi(t){dt\over t}.NEWLINE\]