Positivity of certain functions associated with analysis on elliptic surfaces

In this paper, we study functions of one variable that are called boundary terms of two-dimensional zeta integrals established in recent works of Ivan Fesenkoʼs two-dimensional adelic analysis attached to arithmetic elliptic surfaces. It is known that the positivity of the fourth log derivatives of boundary terms around the origin is a sufficient condition for the Riemann hypothesis of Hasse–Weil L-functions of elliptic curves. We show that such positivity is also a necessary condition under some reasonable technical assumptions.