The representation of a number by a quadratic form

• Komaravolu Chandrasekharan
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 281)

Abstract

The arithmetical function r(n), defined as the number of representations of a positive integer n by the form $$% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGUbWdamaaDaaaleaapeGaaGymaaWdaeaapeGaaGOmaaaakiab % gUcaRiaad6gapaWaa0baaSqaa8qacaaIYaaapaqaa8qacaaIYaaaaO % Gaey4kaSIaamOBa8aadaqhaaWcbaWdbiaaiodaa8aabaWdbiaaikda % aaGccqGHRaWkcaWGUbWdamaaDaaaleaapeGaaGinaaWdaeaapeGaaG % Omaaaaaaa!4432! n_1^2 + n_2^2 + n_3^2 + n_4^2$$ where the (n k ) are integers, is related to the fourth power of the theta-function θ3(0, z), which is defined by the theta-series
$$% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaqa % aaaaaaaaWdbiaadghapaWaaWbaaSqabeaapeGaamOBa8aadaqhaaad % baWdbiaaigdaa8aabaWdbiaaikdaaaWccqGHRaWkcaWGUbWdamaaDa % aameaapeGaaGOmaaWdaeaapeGaaGOmaaaaliabgUcaRiaad6gapaWa % a0baaWqaa8qacaaIZaaapaqaa8qacaaIYaaaaSGaey4kaSIaamOBa8 % aadaqhaaadbaWdbiaaisdaa8aabaWdbiaaikdaaaaaaOGaaiilaiaa % ywW7caWGXbGaeyypa0ZaaWbaaSqabeaacaWGLbGaeqiWdaNaamyAai % aadQhaaaGccaGGSaGaaGzbVlGacMeacaGGTbGaamOEaiabg6da+iaa % icdacaGGSaaal8aabaWdbiaad6gapaWaaSbaaWqaa8qacaaIXaaapa % qabaWcpeGaaiilaiaad6gapaWaaSbaaWqaa8qacaaIYaaapaqabaWc % peGaaiilaiaad6gapaWaaSbaaWqaa8qacaaIZaaapaqabaWcpeGaai % ilaiaad6gapaWaaSbaaWqaa8qacaaI0aaapaqabaWcpeGaeyypa0Ja % eyOeI0IaeyOhIukapaqaaiabg6HiLcqdcqGHris5aOGaafiiaaaa!68D1! \sum\limits_{{n_1},{n_2},{n_3},{n_4} = - \infty }^\infty {{q^{n_1^2 + n_2^2 + n_3^2 + n_4^2}},\quad q{ = ^{e\pi iz}},\quad \operatorname{Im} z > 0,} {\text{ }}$$
, by the formula
$$% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaH4oqCpaWaa0baaSqaa8qacaaIZaaapaqaa8qacaaI0aaaaOWa % aeWaa8aabaWdbiaaicdacaGGSaGaamOEaaGaayjkaiaawMcaaiabg2 % da9maaqahapaqaa8qacaWGYbWaaeWaa8aabaWdbiaad6gaaiaawIca % caGLPaaacaWGXbWdamaaCaaaleqabaWdbiaad6gaaaGccaGGSaGaam % OCamaabmaapaqaa8qacaaIWaaacaGLOaGaayzkaaGaeyypa0JaaGym % aiaacYcacaWGXbGaeyypa0Jaamyza8aadaahaaWcbeqaa8qacqaHap % aCcaWGPbGaamOEaaaakiaacYcaciGGjbGaaiyBaiaadQhacqGH+aGp % caaIWaaal8aabaWdbiaad6gacqGH9aqpcaaIWaaapaqaa8qacqGHEi % sPa0GaeyyeIuoaaaa!5DF6! \theta _3^4\left( {0,z} \right) = \sum\limits_{n = 0}^\infty {r\left( n \right){q^n},r\left( 0 \right) = 1,q = {e^{\pi iz}},\operatorname{Im} z > 0}$$
(cf. (2.1), Ch. X) which enabled us to determine r(n) in terms of the divisors of n.We now consider the more general problem of finding the number of representations of a positive interger by a positive-definite quadratic form, by constructing more general theta-series, and studying their behaviour under modular transformations.

Keywords

Quadratic Form Modular Form Eisenstein Series Chapter VIII Modular Transformation
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