bibkey: nist2026criticalzeros authors: NIST Digital Library of Mathematical Functions year: 2026 title: “DLMF 25.10(i) and 27.4: critical-line zeros and the von Mangoldt Dirichlet series” doi: null url: https://dlmf.nist.gov/25.10.i claim: The source records infinitely many critical-line zeros and the classical von Mangoldt series for the logarithmic derivative of zeta. Neither fact supplies a signed lower comparison for the theta operator. strata_touched: [] license: citation-only triage: anchor

Critical-line infinitude and the complete prime-power series

The inspected DLMF 25.10(i) states that the real $Z(t)$ changes sign infinitely often, hence $\zeta (1/2+it)$ has infinitely many real ordinates. Conjugation symmetry, and isolated zeros give infinitely many distinct positive ordinates. The section cites Titchmarsh, second edition, Section 4.17. This classical infinitude is reused without a proportion estimate or new zero computation.

DLMF 27.4.12 gives ${\sum }_{n\ge 1}\Lambda (n)n^{-z}=-{\zeta }^{\prime }(z)/\zeta (z)$. The absolutely convergent Dirichlet-series domain is $\Re z>1$; $\Lambda (n)\le \log n$ also directly bounds its absolute convergence there. The inspected TeX encoding is 27.4.E12.tex. Every prime power is included, not just the primary primes.

The inspected DLMF release is 1.2.8, released 15 September 2026. These are source statements, without independent proof reproduction or Lean certification. Their full-prime source-range application concerns a precise stronger factorization requirement, not RH or the sign of the original arithmetic form.