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

Abstract

Yang introduced a tempered xi function $\widehatξ(s)$ 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 $\widehatξ$ lies on the critical line $\Re s=1/2$. We disprove this conjecture. After the change of variables $x=e^{2t}$, one has \[ \widehatξ\!\left(\tfrac12+iz\right)=iS(z), \qquad S(z)=\int_0^\infty K(t)\sin(zt)\,dt, \] where $K$ is positive, smooth, and doubly exponentially decaying. Two integrations by parts give $S(y)=K(0)/y+O(y^{-2})$ for real $|y|\to\infty$, 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 $S(y)\to0$ as $y\to\pm\infty$. Thus $S$ has infinitely many nonreal zeros, and consequently $\widehatξ$ 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.