# Landsberg-Schaar relation

The Landsberg-Schaar relation states that for any positive integers $p$ and $q$:

$$\frac{1}{\sqrt{p}}\sum _{n=0}^{p-1}\mathrm{exp}\left(\frac{2\pi i{n}^{2}q}{p}\right)=\frac{{e}^{\pi i/4}}{\sqrt{2q}}\sum _{n=0}^{2q-1}\mathrm{exp}\left(-\frac{\pi i{n}^{2}p}{2q}\right)$$ | (1) |

Although both sides of (1) are mere finite sums,
no one has yet found a proof which uses no infinite^{}
limiting process. One way to prove it is to put
$\tau =2iq/p+\u03f5$, where $\u03f5>0$, in
this identity^{} due to Jacobi:

$$\sum _{n=-\mathrm{\infty}}^{+\mathrm{\infty}}{e}^{-\pi {n}^{2}\tau}=\frac{1}{\sqrt{\tau}}\sum _{n=-\mathrm{\infty}}^{+\mathrm{\infty}}{e}^{-\pi {n}^{2}/\tau}$$ | (2) |

and let $\u03f5\to 0$. The details can be found here (http://planetmath.org/ProofOfJacobisIdentityForVarthetaFunctions). The identity (2) is a basic one in the theory of
theta functions^{}. It is sometimes called the functional equation for the Riemann theta function^{}. See e.g. [2 VII.6.2].

If we just let $q=1$ in the Landsberg-Schaar identity, it reduces to a formula^{}
for the quadratic Gauss sum mod $p$; notice that $p$ need not be prime.

References:

[1] H. Dym and H.P. McKean. *Fourier Series and Integrals*. Academic Press, 1972.

[2] J.-P. Serre. *A Course in Arithmetic*. Springer, 1970.

Title | Landsberg-Schaar relation |
---|---|

Canonical name | LandsbergSchaarRelation |

Date of creation | 2013-03-22 13:23:20 |

Last modified on | 2013-03-22 13:23:20 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 8 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 11L05 |

Synonym | Schaar’s identity |

Related topic | RiemannThetaFunction |