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
, by the formula
(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.
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© 1985 Springer-Verlag Berlin Heidelberg
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Chandrasekharan, K. (1985). The representation of a number by a quadratic form. In: Elliptic Functions. Grundlehren der mathematischen Wissenschaften, vol 281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-52244-4_11
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DOI: https://doi.org/10.1007/978-3-642-52244-4_11
Publisher Name: Springer, Berlin, Heidelberg
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