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Date: Tue, 15 Sep 87 17:59:40 EDT
From: Hugh_L._Montgomery@ub.cc.umich.edu
Subject: Isolated zero of a Dirichlet series

John Thompson wrote me asking the following question:  Does there exist a
Dirichlet series such that it has precisely one zero in its open half-
plane of convergence?  The answer is yes, 1/zeta(s) being an example,
if RH is true, or even if theta = sup Re rho < 1.  Of course if a Dirichlet
series has a zero in its open half-plane of absolute convergence then it
has infinitely many such, so if there is a unique zero as sought then it
must lie in the strip of conditional convergence.  Perhaps there is a
naturally-occurring example which is independent of RH, or perhaps one
can construct something artificial.  Any ideas?
            --  Hugh L. Montgomery,  University of Michigan