Enantioselective access to (−)-indolizidines 167B, 209D, 239AB, 195B and (−)-monomorine from a common chiral synthon

An enantioselective access to (?)-indolizidine alkaloids 167B, 209D, 239AB, 195B and (?)-monomorine from a new chiral synthon is described. The use of (S)-3-(Cbz-amino)-4-(tert-butyldimethylsilyloxy)butanal, obtained from l-aspartic acid, has provided efficient access of the indolizidine frame work through a Horner–Wadsworth–Emmons reaction and reductive cyclization as the key steps.     

A modular look at Farkas’ identities

Recently H. Farkas introduced a new simple arithmetic function and found an identity which involves this function. It is immediate to rewrite this identity as an identity between modular forms and reprove it in this way. We discuss natural generalizations of Farkas’ identity. Surprisingly, in a certain sense, there is only one identity which is an exact analogue of that found by Farkas. At the same time, we present a way to produce infinitely many similar identities. As an application, we obtain a result on non-vanishing of the central critical value of L-functions associated to a cusp Hecke eigenform. Supported by NSF grant DMS-0700933.