اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدCM Evaluations of the Goswami-Sun Series
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In recent work, Sun constructed two $q$-series, and he showed that their limits as $qrightarrow1$ give new derivations of the Riemann-zeta values $zeta(2)=pi^2/6$ and $zeta(4)=pi^4/90$. Goswami extended these series to an infinite family of $q$-series, which he analogously used to obtain new derivations of the evaluations of $zeta(2k)inmathbb{Q}cdotpi^{2k}$ for every positive integer $k$. Since it is well known that $Gammaleft(frac{1}{2}right)=sqrt{pi}$, it is natural to seek further specializations of these series which involve special values of the $Gamma$-function. Thanks to the theory of complex multiplication, we show that the values of these series at all CM points $tau$, where $q:=e^{2pi itau}$, are algebraic multiples of specific ratios of $Gamma$-values. In particular, classical formulas of Ramanujan allow us to explicitly evaluate these series as algebraic multiples of powers of $Gammaleft(frac{1}{4}right)^4/pi^3$ when $q=e^{-pi}$, $e^{-2pi}$.
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