Infinitely Many Off-Critical-Line Zeros of the Tempered Xi Function: A Disproof of Yang's Conjecture

Abstract

Yang introduced a tempered xi function ξ̂(s) by replacing the hyperbolic cosine in a classical integral representation of the Riemann xi function by a hyperbolic sine, and conjectured that every zero of ξ̂ lies on the critical line ℜ s=1/2. We disprove this conjecture. After the change of variables x=e^2t, one has ξ̂ (1/2+iz)=iS(z), S(z)=∫₀^∞ K(t)sin(zt) dt, where K is positive, smooth, and doubly exponentially decaying. Two integrations by parts give S(y)=K(0)/y+O(y⁻²) for real |y|→∞, with K(0)>0; hence S has only finitely many real zeros. On the other hand S is an entire function of order at most one. If it had only finitely many zeros in the complex plane, Hadamard factorization would force S(z)=P(z)e^az+b, which is incompatible with S(y)→0 as y→±∞. Thus S has infinitely many nonreal zeros, and consequently ξ̂ has infinitely many zeros off the critical line. The proof is unconditional and does not use the Riemann Hypothesis or numerical zero finding. A short numerical illustration is included.

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BibTeX RIS

BibTeX

@misc{kazin2026infinitely,
  title = {{Infinitely Many Off-Critical-Line Zeros of the Tempered Xi Function: A Disproof of Yang's Conjecture}},
  author = {Aiken Kazin and Shirali Kadyrov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29898},
  eprint = {2609.29898},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29898},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Infinitely Many Off-Critical-Line Zeros of the Tempered Xi Function: A Disproof of Yang's Conjecture
AU  - Aiken Kazin
AU  - Shirali Kadyrov
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29898
UR  - https://arxiv.org/abs/2609.29898
AB  - Yang introduced a tempered xi function ξ̂(s) by replacing the hyperbolic cosine in a classical integral representation of the Riemann xi function by a hyperbolic sine, and conjectured that every zero of ξ̂ lies on the critical line ℜ s=1/2. We disprove this conjecture. After the change of variables x=e^2t, one has ξ̂ (1/2+iz)=iS(z), S(z)=∫_0^∞ K(t)sin(zt) dt, where K is positive, smooth, and doubly exponentially decaying. Two integrations by parts give S(y)=K(0)/y+O(y^−2) for real |y|→∞, with K(0)>0; hence S has only finitely many real zeros. On the other hand S is an entire function of order at most one. If it had only finitely many zeros in the complex plane, Hadamard factorization would force S(z)=P(z)e^az+b, which is incompatible with S(y)→0 as y→±∞. Thus S has infinitely many nonreal zeros, and consequently ξ̂ has infinitely many zeros off the critical line. The proof is unconditional and does not use the Riemann Hypothesis or numerical zero finding. A short numerical illustration is included.
ER  -

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