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We obtain lower bounds of the correct order of magnitude for the 2 k th moment of the Riemann zeta function for all  k ≥1. Previously such lower bounds were known only for rational values of  k , with the bounds depending on the height of the rational number  k . Our new bounds are continuous in  k , and thus extend also to the case when  k  is irrational. The method is a refinement of an approach of Rudnick and Soundararajan, and applies also to moments of  L ‐functions in families.

CONTINUOUS LOWER BOUNDS FOR MOMENTS OF ZETA AND L ‐FUNCTIONS