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Schaar's Identity


A generalization of the Gaussian sum. For p and q of opposite parity (i.e., one is even and the other is odd), Schaar's identity states

 1/(sqrt(q))sum_(r=0)^(q-1)e^(-piir^2p/q)=(e^(-pii/4))/(sqrt(p))sum_(r=0)^(p-1)e^(piir^2q/p).

Schaar's identity can also be written so as to be valid for p, q with pq even.


See also

Gaussian Sum

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References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Evans, R. and Berndt, B. "The Determination of Gauss Sums." Bull. Amer. Math. Soc. 5, 107-129, 1981.

Referenced on Wolfram|Alpha

Schaar's Identity

Cite this as:

Weisstein, Eric W. "Schaar's Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchaarsIdentity.html

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