Lemma Huanum

Lemma Huanum,^[1] ex Hua Loo-keng appellatum, in mathematica est aestimatio summarum exponentialium. Quae dicit si res P sit polynomialis per integrales aestimatus^? gradús k, ${\displaystyle \varepsilon }$ sit numerus realis positivus, et f sit functio realis per

${\displaystyle f(\alpha )=\sum _{x=1}^{N}\exp(2\pi iP(x)\alpha )}$

definita, tum

${\displaystyle \int _{0}^{1}|f(\alpha )|^{\lambda }d\alpha \ll _{P,\varepsilon }N^{\mu (\lambda )}}$,

ubi ${\displaystyle (\lambda ,\mu (\lambda ))}$ in linea polygonali iacet cum verticibus

${\displaystyle (2^{\nu },2^{\nu }-\nu +\varepsilon ),\quad \nu =1,\ldots ,k.}$

Notae

1. Hua, Loo-keng. 1938. On Waring's Problem. Quarterly Journal of Mathematics 9(1):199–202. doi 10.1093/qmath/os-9.1.199.

mathematica

Haec stipula ad mathematicam spectat. Amplifica, si potes!