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Monday, September 26, 2016

Entry 23

For powers k=4n+3, Ramanujan gave n=1n3e2nπ1=13e2π1+23e4π1+33e6π1+=Γ(14)82105π61240 n=1n7e2nπ1=17e2π1+27e4π1+37e6π1+=3Γ(14)162175π121480 n=1n11e2nπ1=111e2π1+211e4π1+311e6π1+=189Γ(14)24222513π1869165520 while for k=4n+1 it evaluates to a rational number n=1n5e2nπ1=15e2π1+25e4π1+35e6π1+=1504 n=1n9e2nπ1=19e2π1+29e4π1+39e6π1+=1264 n=1n13e2nπ1=113e2π1+213e4π1+313e6π1+=124 and so on. The case k=1 diverges.

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