Let $(\frac{m}{n})$ denote the usual quadratic Jacobi symbol. What is the abscissa of convergence of the double Dirichlet series ? $$ \sum_{\substack{m,n \in \mathbb{N} \\ \gcd(m,n)=1 \\m,n\equiv 1 \mod{4}}} \left(\frac{m}{n}\right)(mn)^{-s} $$
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1$\begingroup$ In what sense is convergence intended? That is how do $m$ and $n$ go to infinity? $\endgroup$– LuciaFeb 26, 2015 at 18:43
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$\begingroup$ I do not mind about this at all. I guess $M\leq M,n\leq N$ and $M,N \to \infty$ looks natural but if one can get a smaller abscissa for $m,n\leq x,x \to \infty$ or perhaps $mn\leq x,x \to \infty$ then I would also be happy with this. $\endgroup$– Dr. PiFeb 26, 2015 at 21:36
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