The Yousef L-Function: Corrections, the General Variable s, and the Generalized Redheffer Matrix

We correct several numerical errors in the definition of the Yousef L-function

that appeared in an earlier preprint. We provide the correct values of Q(p) for odd

primes p ≤ 39, derive the corrected Euler product, and present the general formula

for the difference function D(s) = L(s, χ4) − LYousef(s) at an arbitrary complex

variable s. The coefficient c(n) = χ4(n)−G(n) takes values in {0, ±2, ±(1+i), ±(1−

i)}. After extracting the factor 2, we define Z(n) = 2/c(n) and prove the structural

identity ℜ(Z(n)) = χ4(n), valid for all s. This reveals that the non-periodicity of

the Yousef L-function is confined to the imaginary part of Z(n). We also establish

a generalized Redheffer determinant formula and provide a Mellin representation of

D(s) for all s with ℜ(s) > 1.