A new four parameter q-series identity and its partition implications

We prove a new four parameter q-hypergeometric series identity from which the three parameter key identity for the Goellnitz theorem due to Alladi, Andrews, and Gordon, follows as a special case by setting one of the parameters equal to 0. The new identity is equivalent to a four parameter partition theorem which extends the deep theorem of Goellnitz and thereby settles a problem raised by Andrews thirty years ago. Some consequences including a quadruple product extension of Jacobi's triple product identity, and prospects of future research are briefly discussed.Comment: 25 pages, in Sec. 3 Table 1 is added, discussion is added at the end of Sec. 5, minor stylistic changes, typos eliminated. To appear in Inventiones Mathematica

Asymptotic estimates of sums involving the Moebius function

Let p(n) denote the smallest prime factor of an integer n>1 and let p(1)=[infinity]. We study the asymptotic behavior of the sum M(x,y)=[Sigma]1nx,p(n)>y[mu](n) and use this to estimate the size of A(x)=max|f||[Sigma]2nx[mu](n)f(p(n))|, where [mu](n) is the Moebius function. Applications of bounds for A(x), M(x,y) and similar quantities are discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24065/1/0000317.pd