First Hardy–Littlewood zeta-function conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\zeta(s)$ denote the meromorphic continuation to $\mathbb C$ of $\sum_{n\ge1}n^{-s}$, initially defined for $\operatorname{Re}s>1$. A real ordinate $t$ is an odd-order zero on the critical line if $\zeta(\tfrac12+it)=0$ and the multiplicity of this zero as a zero of the holomorphic function $\zeta$ is an odd positive integer.

Statement recorded in the supplied source.

Is it true that for every $\varepsilon>0$ there exists $T_0(\varepsilon)>0$ such that for every real $T\ge T_0(\varepsilon)$ the interval \[\left(T,T+T^{1/4+\varepsilon}\right]\] contains an ordinate of an odd-order zero on the critical line? [S2]

Short English statement

For every positive allowance in the exponent, must every sufficiently high interval of the specified length contain a critical-line zero of odd multiplicity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400064 (AMR-093-0064). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, First Hardy–Littlewood zeta-function conjecture. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 Wikipedia contributors, Hardy–Littlewood zeta function conjectures, revision 1229600719 (2026) First conjecture. https://en.wikipedia.org/w/index.php?title=Hardy%E2%80%93Littlewood+zeta+function+conjectures&oldid=1229600719.