Eisenstein’s Proofs Using Complex Analysis

Oswald Baumgart

Chapter Chapter 11 in The Quadratic Reciprocity Law, 2015, pp 107-109 from Springer

Abstract: Abstract If r represents a half-system modulo q, then so does rp. Setting $$\mathit{rp} \equiv \varepsilon r^{{\prime}}\bmod q$$ , where $$\varepsilon = \pm 1$$ , and where r ′ belongs to the same half-system as r, then for an arbitrary integer ω we have $$\displaystyle{\frac{\mathit{pr}\omega } {q} \equiv \frac{\varepsilon r^{{\prime}}\omega } {q} \bmod \omega.}$$ This implies $$\displaystyle{P\Big(\frac{\mathit{pr}\omega } {q} \Big) = P\Big(\frac{\varepsilon r^{{\prime}}\omega } {q} \Big),}$$ where P denotes any simply periodic function with period ω.

Keywords: Number Theory; Periodic Function; Complex Analysis; Theoretic Method; Sine Function (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-16283-6_11

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