math.NT — Number Theory

A counterexample to Haglund's monotonic-zeros conjecture for the Riemann $\Xi$ approximants

Contributed by Kunal Tyagi

Haglund introduced the partial sums $\Xi_N = \sum_{n \le N} \Phi_n$ of Riemann's expansion of the $\Xi$-function in incomplete gamma functions and conjectured that for every $N$ the zeros of $\Xi_N$ in the closed first quadrant, listed by increasing real part, have nondecreasing imaginary parts. The conjecture implies the Riemann hypothesis; Haglund's computations supported a weak form of it for $N \le 10$. We show that the conjecture is false for $N = 27$: $\Xi_{27}$ has a real zero in $(3144.8946, 3144.8947)$ and a non-real zero $z_0 = 3143.2206824215\ldots + 0.3152587993\ldots\, i$ in the first quadrant, whose real part is smaller. The weak form fails as well. Both facts are certified twice, independently, in interval arithmetic (Arb ball arithmetic; outward-rounded intervals with proved truncation bounds), and $z_0$ is enclosed in a square of half-width $10^{-48}$. The same violation occurs at $N = 24$ for the companion approximants $\xi_N = \Xi_N + c_N$. The failure has a mechanism: on the critical line $\Xi_N < \Xi < \xi_N$, so the real zeros of $\Xi_N$ lie in positive lobes of $\Xi$ that rise above a positive level, and a low lobe followed by a higher one near the height $4(N+1)^2$, where the approximant departs from $\Xi$, can leave a non-real zero below real zeros. The same structure shows that the number of positive real zeros of $\Xi_N$, counted with multiplicity, is odd, and that $x^2\Xi_N(x)$ tends to a negative limit on the real axis for every $N$.