Joint extreme values of L-functions

Abstract

We consider L-functions \(L_1,\ldots ,L_k\) from the Selberg class which have polynomial Euler product and satisfy Selberg’s orthonormality condition. We show that on every vertical line \(s=\sigma +it\) with \(\sigma \in (1/2,1)\), these L-functions simultaneously take large values of size \(\exp \left( c\frac{(\log t)^{1-\sigma }}{\log \log t}\right) \) inside a small neighborhood. Our method extends to \(\sigma =1\) unconditionally, and to \(\sigma =1/2\) on the generalized Riemann hypothesis. We also obtain similar joint omega results for arguments of the given L-functions.