The Fejér-Dirichlet Lift: Entire Functions and $\hat{I}\P$-Factorization Identities

Abstract

A Fejér-Dirichlet lift is developed that turns divisor information at the integers into entire interpolants with explicit Dirichlet-series factorizations. For absolutely summable weights the lift interpolates $(a\ast 1)(n)$ at each integer $n$ and has Dirichlet series $\hat{I}\P (s)A(s)$ on $\Re s>1$. Two applications are emphasized. First, for $q>1$ an entire function $\mathfrak{F}(\cdot ,q)$ is constructed that vanishes at primes and is positive at composite integers; a tangent-matched variant ${\mathfrak{F}}^{♯}$ is shown to admit an explicit, effective threshold $P_0(q)$ such that for every odd prime $p\ge P_0(q)$ the interval $(p-1,p)$ is free of real zeros and $x=p$ is a boundary zero of multiplicity two. Second, a renormalized lift for $a=\hat{I}¼\ast \hat{I}›$ produces an entire interpolant of $\hat{I}›(n)$ and provides a constructive viewpoint on the appearance of $\hat{I}{\P }^{\prime }(s)/\hat{I}\P (s)$ through the FD-lift spectrum. A Polylog-Zeta factorization for the geometric-weight case links $\hat{I}\P (s)$ with ${\mathrm{Li}}_s(1/q)$. All prime/composite statements concern integer arguments. Scripts reproducing figures and numerical checks are provided in a public repository with an archival snapshot.

58 pages, 13 figures