Exponential Series Defined by Sequences and Their Connections with the Riemann Zeta Function
Description
We study exponential series associated with real sequences,
\[
\chi_{c_n}(s)=\sum_{n=1}^{\infty} e^{-s c_n},
\]
where $(c_n)_{n\in\mathbb{N}}$ is a sequence of real numbers and $s\in\mathbb{C}$.
We establish convergence criteria for these series in terms of the asymptotic parameter
\[
L=\lim_{n\to\infty}\frac{\ln n}{c_n},
\]
showing that $L$ completely determines the convergence threshold of $\chi_{c_n}(s)$.
When $0<L<\infty$, we prove that the corresponding exponential series is asymptotically governed by the scaled Riemann zeta function
\[
\zeta_L(s)=\zeta\!\left(\frac{s}{L}\right).
\]
More precisely, after modifying finitely many terms if necessary, $\chi_{c_n}(s)$ can be sandwiched between arbitrarily close scaled zeta functions and approximates $\zeta_L(s)$ arbitrarily well in the half-plane $\Re(s)>L$.
We also investigate the polynomial-growth family $c_n=n^p$ ($p>0$), 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.