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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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Komaravolu Chandrasekharan
    • 1
  1. 1.Eidgenössische Technische Hochschule ZürichZürichSwitzerland

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