Completing the Square

Wei Zhang, Columbia University
Fine Hall 314

L-functions (e.g., Riemann zeta function) constitute a special class of functions in one complex variable. It is “natural" to look at the Taylor expansion of L-functions (suitably normalized) at their “centers”. Assuming one of the standard conjectures on L-functions (i.e., the generalized Riemann hypothesis), I will explain how to deduce that all Taylor coefficients are positive (if they are all real). How can one prove the positivity unconditionally? A natural idea is by completing the square. We present some successful examples where the “square roots” bear interesting geometric meanings.