The Fejér-Dirichlet Lift: Entire Functions and $ζ$-Factorization Identities
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915495933902848 |
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| author | Fuchs, Sebastian |
| author_facet | Fuchs, Sebastian |
| contents | 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 $(a*1)(n)$ at each integer $n$ and has Dirichlet series $ζ(s)A(s)$ 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 $\mathfrak F^{\sharp}$ 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 $a=μ*Λ$ produces an entire interpolant of $Λ(n)$ and provides a constructive viewpoint on the appearance of $ζ'(s)/ζ(s)$ through the FD-lift spectrum. A Polylog-Zeta factorization for the geometric-weight case links $ζ(s)$ with $\operatorname{Li}_s(1/q)$. All prime/composite statements concern integer arguments. Scripts reproducing figures and numerical checks are provided in a public repository with an archival snapshot. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_12297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Fejér-Dirichlet Lift: Entire Functions and $ζ$-Factorization Identities Fuchs, Sebastian General Mathematics 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 $(a*1)(n)$ at each integer $n$ and has Dirichlet series $ζ(s)A(s)$ 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 $\mathfrak F^{\sharp}$ 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 $a=μ*Λ$ produces an entire interpolant of $Λ(n)$ and provides a constructive viewpoint on the appearance of $ζ'(s)/ζ(s)$ through the FD-lift spectrum. A Polylog-Zeta factorization for the geometric-weight case links $ζ(s)$ with $\operatorname{Li}_s(1/q)$. All prime/composite statements concern integer arguments. Scripts reproducing figures and numerical checks are provided in a public repository with an archival snapshot. |
| title | The Fejér-Dirichlet Lift: Entire Functions and $ζ$-Factorization Identities |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2509.12297 |