Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), $P_{[p/6]}\left(t\right)\equiv -(3/p){\sum }_{x=0}^{p-1}((x³-3x+2t)/p)\left(modp\right)$ and $\left({\sum }_{x=0}^{p-1}((x³+mx+n)/p)\right)²\equiv ((-3m)/p){\sum }_{k=0}^{[p/6]}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\left(\genfrac{}{}{0pt}{}{3k}{k}\right)\left(\genfrac{}{}{0pt}{}{6k}{3k}\right){((4m³+27n²)/\left(12³·4m³\right))}^k\left(modp\right)$, where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\left(\genfrac{}{}{0pt}{}{3k}{k}\right)\left(\genfrac{}{}{0pt}{}{6k}{3k}\right)/m^k\left(modp²\right)$, where m is an integer not divisible by p.

La formule généralisant la loi de réciprocité quadratique de Legendre et exprimant le reste par huit de la signature d'une forme quadratique entière non dégénérée à l'aide d'une somme de Gauss est attribuée par Milnor à Milgram, la faisant remonter à Braun. Le formalisme de Witt la réduit au cas de dimension 1 que Chandrasekharan attribue à Cauchy et Kronecker. Braun soulignait que les preuves de ces formules nécessitent des moyens d'analyse. Une propriété métrique de l'octogone...

We prove analogue statements of the spherical maximal theorem of E. M. Stein, for the lattice points Zn. We decompose the discrete spherical measures as an integral of Gaussian kernels st,ε(x) = e2πi|x|2(t + iε). By using Minkowski's integral inequality it is enough to prove Lp-bounds for the corresponding convolution operators. The proof is then based on L2-estimates by analysing the Fourier transforms ^st,ε(ξ), which can be handled by making use of the circle method for exponential sums. As a...