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The representation of a number as a sum of four squares

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

Abstract

We have seen in Chapter VIII that the identity
$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaH3oaAdaqadaWdaeaapeGaamOEaaGaayjkaiaawMcaaiabg2da % 9iaadwgapaWaaWbaaSqabeaapeGaeqiWdaNaamyAaiaadQhacaGGVa % GaaGymaiaaikdaaaGccqaH4oqCpaWaaSbaaSqaa8qacaaIZaaapaqa % baGcpeWaaeWaa8aabaWdbmaalaaapaqaa8qacaaIXaaapaqaa8qaca % aIYaaaaiabgUcaRmaalaaapaqaa8qacaWG6baapaqaa8qacaaIYaaa % aiaacYcacaaIZaGaamOEaaGaayjkaiaawMcaaiaacYcaciGGjbGaai % yBaiaadQhacqGH+aGpcaaIWaaaaa!535D! \eta \left( z \right) = {e^{\pi iz/12}}{\theta _3}\left( {\frac{1}{2} + \frac{z}{2},3z} \right),\operatorname{Im} z > 0 $$
(1.1)
, which connects Dedekind’s η-function with the theta-function 6 3 implies Euler’s theorem on pentagonal numbers. That was proved by analytical methods in two different ways. The first consisted in representing θ3(υ, z), initially defined by an infinite series, as an infinite product, and identifying the defining product of η(z) with that which results from the right-hand side of (1.1). The second consisted in combining the transformation formula for θ3(υ, z), namely
$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaH4oqCpaWaaSbaaSqaa8qacaaIZaaapaqabaGcpeWaaeWaa8aa % baWdbiaaicdacaGGSaGaeyOeI0YaaSaaa8aabaWdbiaaigdaa8aaba % WdbiaaikdaaaaacaGLOaGaayzkaaGaeyypa0ZaaOaaa8aabaWdbmaa % laaapaqaa8qacaWG6baapaqaa8qacaWGPbaaaiaac6caaSqabaGccq % aH4oqCpaWaaSbaaSqaa8qacaaIZaaapaqabaGcpeWaaeWaa8aabaWd % biaaicdacaGGSaGaamOEaaGaayjkaiaawMcaaiaacYcaciGGjbGaai % yBaiaadQhacqGH+aGpcaaIWaaaaa!4F1F! {\theta _3}\left( {0, - \frac{1}{2}} \right) = \sqrt {\frac{z}{i}.} {\theta _3}\left( {0,z} \right),\operatorname{Im} z > 0 $$
(1.2)
, with the functional equation of η(z), namely
$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaH3oaAdaqadaWdaeaapeGaeyOeI0YaaSaaa8aabaWdbiaaigda % a8aabaWdbiaaikdaaaaacaGLOaGaayzkaaGaeyypa0ZaaOaaa8aaba % Wdbmaalaaapaqaa8qacaWG6baapaqaa8qacaWGPbaaaiabeE7aOnaa % bmaapaqaa8qacaWG6baacaGLOaGaayzkaaaaleqaaOGaaiilaiGacM % eacaGGTbGaamOEaiabg6da+iaaicdaaaa!4923! \eta \left( { - \frac{1}{2}} \right) = \sqrt {\frac{z}{i}\eta \left( z \right)} ,\operatorname{Im} z > 0 $$
(1.3)
, (1.3) so as to construct a modular function which vanishes identically in the upper half-plane Im z>0, and thereby yields (1.1).

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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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