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Throughout, $p$ will denote a prime integer, and $k$ an arbitrary integer. I have worked through V. Lebesgue's proof of quadratic reciprocity outlined by Keith Conrad in this MO thread, and I feel ...

I'm referring to this proof. The key formula ("Eisenstein's Lemma") is $$\left(\frac{q}{p}\right)=(-1)^{\sum_{u}\lfloor\frac{qu}{p}\rfloor},\text{ where $u=2,4,\ldots,p-1$}$$ The sum in the ...

In Samuel James Patterson's article titled Gauss Sums in The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, Patterson says "Hecke [proved] a beautiful theorem on the different ...

We have explicit expressions for the quadratic Hilbert symbol over $\mathbb Q$, for example $\left(\dfrac{x,y}2\right)_2=(-1)^{\frac{x-1}2\frac{y-1}2} (x,y\ne2)$. Are similar expressions known for the ...

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 \...