Exponential Series Defined by Sequences and Their Connections with the Riemann Zeta Function

We study exponential series associated with real sequences, \[ χc_n(s)=∑ₙ₌₁∞ e-s c_n, \] where (cₙ)ₙ is a sequence of real numbers and s. We establish convergence criteria for these series in terms of the asymptotic parameter \[ L=limn→∞ln n/c_n, \] showing that L completely determines the convergence threshold of χcₙ(s). When $0 L$. We also investigate the polynomial-growth family cₙ=nᵖ ($p>0$), which corresponds to the complementary regime $L=0$. For this family, we obtain alternative integral representations, establish zero-free regions, and prove the existence of arbitrarily small values near the imaginary axis. Together, these two regimes provide a unified framework connecting exponential series defined by sequences with the asymptotic behavior of the Riemann zeta function.