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 S₍, ₃ (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 D₂ₑ (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) /2ˢ, 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, and F (4, 2) =41π⁴/5760. We further establish a general Hurwitz zeta representation and prove that F (s, d) interpolates monotonically between η (s) and ζ (s) as d varies.