Block-Alternating Weierstrass Products and Dirichlet-Type Series: A Structural Connection via Floor-Function Deformation
Year: 2026
Type: preprint
Abstract: We study a floor-function block-alternating deformation of the classical sinc product and identify a natural connection with a family of block-alternating Dirichlet-type series. The deformed sinc product S_{n,d}(x) is defined by imposing a sign pattern σ(m,d):=(−1)^{⌊(m−1)/d⌋} on the Weierstrass factors, introducing a discrete block parameter d. Its logarithmic expansion coefficients D_{2r}(d) coincide, by a direct identification, with the block-alternating Dirichlet-type series F(s,d)=Σ(−1)^{⌊(n−1)/d⌋}n^{−s}. This unifying perspective connects infinite-product expansions and Dirichlet series through a single discrete parameter. The main analytic contribution is the closed form for d=2: F(s,2)=β(s)+η(s)/2^s, where β and η denote the Dirichlet beta and eta functions. This yields explicit special values including F(2,2)=G+π²/48, where G is Catalan's constant. We further establish a general Hurwitz zeta representation and prove that F(s,d) lies between η(s) and ζ(s), with F(s,1)=η(s) and F(s,d)→ζ(s) as d→∞. Note (v2): The special value F(4,2) has been corrected. In v1, β(4) was incorrectly replaced by 5π⁴/768. The correct value is F(4,2) = β(4) + 7π⁴/11520. No other results are affected by this correction.
Author: Fujii, Masanori
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7149911835
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