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Sums of \(L\)-functions over rational function fields. (English) Zbl 1217.11108

Let \({\mathbb F}_q\) be the finite field with \(q\) elements, \({\mathcal O}={\mathbb F}_q[t]\), \({\mathcal O}_{\text{mon}}\) the set of monic polynomials in \({\mathbb F}_q[t]\), \(\zeta_{\mathcal O}(s)=\frac{1}{1-q^{1-s}}\). Let \(n\geq 2\). The authors define the double Dirichlet series \[ Z_1(s,w;\delta_i)=\sum_{{d,m\in {\mathcal O}_{\text{mon}}\atop \deg m\equiv i\pmod n}}\frac{\chi_{m_0}(\hat{d})a(d,m)}{|m|^w|d|^s}, \]
\[ Z_2(s,w;\delta_i)=\zeta_{\mathcal O}(nw-\frac{n}{2}+1)\sum_{{d,m\in {\mathcal O}_{\text{mon}}\atop \deg m\equiv i\pmod n}}\frac{g(1,\varepsilon,\chi_{m_0})}{\sqrt{|m_\flat|}}\frac{\bar{\chi}_{m_0}(\hat{d})b(d,m)}{|m|^w|d|^s}, \] where \(m_0\) is the \(n\)th power free part of \(m\), \(\hat{d}\) is the part of \(d\) relatively prime to \(m_0\), \(\chi_{m_0}(\hat{d})\) is the \(n\)th power residue symbol, \(a(d,m)\) and \(b(d,m)\) are weighing factors, \(g(1,\varepsilon,\chi_{m_0})\) is a Gauss sum, \(m_\flat\) is the product of primes dividing \(m_0\), \(|m|=q^{\deg m}\). They prove the functional equations \[ Z_1(s,w;\delta_i)=\begin{cases} q^{2s-1}\frac{1-q^{-s}}{1-q^{s-1}}Z_2(1-s,w+s-\frac12;\delta_0),&i=0\\ q^{2s-1}q^{\frac12-s}\frac{\bar{\tau}(\epsilon^i)}{\sqrt{q}}Z_2(1-s,w+s-\frac12;\delta_i),&0<i<n, \end{cases} \] with a finite field Gauss sum \(\tau(\varepsilon^i)\). The series \(Z_1\) and \(Z_2\) are rational functions of \(q^{-s}\) and \(q^{-w}\).

MSC:

11R58 Arithmetic theory of algebraic function fields
11F68 Dirichlet series in several complex variables associated to automorphic forms; Weyl group multiple Dirichlet series
11M41 Other Dirichlet series and zeta functions
11T23 Exponential sums
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