The representation of a number as a sum of four squares

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).