Landsberg–Schaar Relation
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In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
and
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
, the Landsberg–Schaar relation (or identity) is the following equation, which is valid for arbitrary positive integers ''p'' and ''q'': : \frac\sum_^\exp\left(\frac\right)= \frac\sum_^\exp\left(-\frac\right). The standard way to prove it is to put  =  + ''ε'', where ''ε'' > 0 in this identity due to
Jacobi Jacobi may refer to: * People with the surname Jacobi (surname), Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenva ...
(which is essentially just a special case of the
Poisson summation formula In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a ...
in classical harmonic analysis): : \sum_^e^=\frac \sum_^e^ and then let ''ε'' â†’ 0. A proof using only finite methods was discovered in 2018 by Ben Moore. If we let ''q'' = 1, the identity reduces to a formula for the
quadratic Gauss sum In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; fo ...
modulo ''p''. The Landsberg–Schaar identity can be rephrased more symmetrically as : \frac\sum_^\exp\left(\frac\right)= \frac\sum_^\exp\left(-\frac\right) provided that we add the hypothesis that ''pq'' is an even number.


References

{{DEFAULTSORT:Landsberg-Schaar relation Theorems in analytic number theory