Block-Alternating Weierstrass Products and Dirichlet-Type Series: A Structural Connection via Floor-Function Deformation

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 Sn,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 D₂ᵣ(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.