Functional Dirichlet Series, Integral Representations of a Generalized Zeta Function, and Applications to Sustainability

Natanael Karjanto (), Romano Emmanuelle and Bharath Sriraman ()
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Natanael Karjanto: Sungkyunkwan University, Department of Mathematics, College of Natural Science
Romano Emmanuelle: Chiba University, Department of Mathematics and Informatics
Bharath Sriraman: University of Montana – Missoula, Department of Mathematical Sciences

A chapter in Handbook of Visual, Experimental and Computational Mathematics, 2026, pp 391-439 from Springer

Abstract: Abstract We investigate a generalized Dirichlet series with functional coefficients involving products of exponential and hyperbolic secant functions, extending our previous work on harmonic series representations. By introducing a sequence of functions f n ( s , x ) $$f_n(s,x)$$ parameterized by a complex number s with Re ( s ) > 1 $$\text{Re}(s) > 1$$ and real number x ≥ 0 $$x \geq 0$$ , we establish both pointwise and uniform convergence properties of the associated series ∑ n = 1 ∞ f n ( s , x ) $$\displaystyle \sum _{n=1}^{\infty } f_n(s,x)$$ . We verify that although the series converges pointwise on the entire domain E = { ( s , x ) : Re ( s ) > 1 , x ≥ 0 } $$E = \{(s,x) : \, \text{Re}(s) > 1, x \geq 0\}$$ , uniform convergence fails on E but holds on appropriately restricted subsets, leading to locally uniform convergence on compact subsets. The sum function is expressed explicitly in terms of polylogarithm functions and the Riemann zeta function. Through integral representations obtained via term-by-term integration, we derive connections to classical results, that is, as s → 1 + $$s \to 1^+$$ , our framework provides an alternative proof of the divergence of the harmonic series, while the case s = 2 $$s = 2$$ recovers Euler’s solution to the Basel problem, ζ ( 2 ) = π 2 ∕ 6 $$\zeta (2) = \pi ^2/6$$ . We characterize convergence rates on restricted domains, showing exponential decay O ( r ε n ) $$O(r_\varepsilon ^n)$$ when x is bounded away from zero and polynomial decay O ( 1 ∕ n 1 + δ ) $$O(1/n^{1+\delta })$$ when Re ( s ) $$\text{Re}(s)$$ is bounded away from the critical value 1. Finally, we explore potential applications to sustainability through a work-rate model illustrating diminishing returns in resource allocation and discuss emerging theoretical connections between the Riemann zeta function and environmental modeling in complex systems, climate dynamics, and extreme event analysis.

Keywords: Functional Dirichlet series; Generalized Riemann zeta function; Integral representations; Polylogarithm functions; Harmonic series; Sustainability applications (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16368-4_91

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