number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

and

harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...

, 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 (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 (mathematics), function to values of the function's continuous Fourier transform. Consequently, the pe ...

in classical harmonic analysis): :

\sum_^e^=\frac
\sum_^e^

and then let ''ε'' → 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; f ...

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