Second 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.

The Riemann zeta function is $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ for $\operatorname{Re}s>1$, continued meromorphically to $\mathbb C$. A zero has odd order if its multiplicity as a zero of this meromorphic function is a positive odd integer. For $T>0$, put \[N_0(T)=\#\{t\in(0,T]:\zeta(1/2+it)=0\text{ and the zero has odd order}\}.\] Thus this counts the distinct positive ordinates of odd-order zeros on the critical line $\operatorname{Re}s=1/2$; it includes all odd multiplicities, rather than only simple zeros.

Statement recorded in the supplied source.

For every $\varepsilon>0$, do there exist constants $T_0(\varepsilon)>0$ and $c(\varepsilon)>0$ such that, for every $T\ge T_0(\varepsilon)$ and $H=T^{1/2+\varepsilon}$, one has $N_0(T+H)-N_0(T)\ge c(\varepsilon)H$? [S2]

Short English statement

Does every sufficiently high interval of the specified length contain at least a constant times its length many odd-order zeros of the Riemann zeta function on its critical line?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400183 (AMR-093-0183). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, Hardy--Littlewood zeta function conjectures, revision 1229600719; source exposition citing G. H. Hardy and J. E. Littlewood, The zeros of Riemann's zeta-function on the critical line, Mathematische Zeitschrift 10 (1921), 283--317, doi:10.1007/BF01211614, Conjectures, definition of $N_0(T)$ and item 2. https://en.wikipedia.org/w/index.php?title=Hardy%E2%80%93Littlewood_zeta_function_conjectures&oldid=1229600719.