Soundararajan Conjecture on Zeta Zero Spacing

Definitions and mathematical statement

Let $\zeta(s)$ be the Riemann zeta function, and assume the Riemann Hypothesis: every nontrivial zero of $\zeta$ has real part $1/2$. Let $\rho'=\beta'+i\gamma'$ run over the nontrivial zeros of $\zeta'(s)$ with positive ordinates tending to infinity. Let $\gamma$ run over the positive ordinates of the nontrivial zeros of $\zeta(s)$, and let $\gamma^+$ denote the next such ordinate in the source's ordering.

Soundararajan's conjecture, as supplied, asks for the equivalence \[ \liminf_{\substack{\zeta'(\beta'+i\gamma')=0\\\gamma'\to\infty}} (\beta'-\tfrac12)\log\gamma'=0 \quad\Longleftrightarrow\quad \liminf_{\gamma\to\infty}(\gamma^+-\gamma)\log\gamma=0. \] The left limit is along the designated derivative zeros, not over all real heights. In a customary ordering $\gamma_1\leq\gamma_2\leq\cdots$ with multiplicity, the right side is \[ \liminf_{n\to\infty}(\gamma_{n+1}-\gamma_n)\log\gamma_n=0. \] The extraction does not expressly specify multiplicity versus strictly distinct ordinates; this must be checked before fixing the successor convention.

The source additionally defines $\gamma_c$ as the ordinate of a zeta zero nearest to $\gamma'$, meaning one minimizing $|\gamma_c-\gamma'|$. It supplies no tie-breaking convention, and $\gamma_c$ does not occur in the displayed equivalence. No additional conclusion involving it is inferred. The intended nontrivial-zero convention for $\zeta'$ should likewise be confirmed; derivative zeros are not silently restricted to $0<\beta'<1$.

Short English statement

Assuming RH, do derivative zeros approach the critical line on the logarithmic scale exactly when consecutive zeta-zero ordinates have arbitrarily small logarithmically normalized gaps? This is the supplied Soundararajan equivalence.

Sources

• Local UnsolvedMath registry, permanent ID $30000940$, source code catalogue entry, "Soundararajan Conjecture on Zeta Zero Spacing." The original statement attributes the conjecture to Soundararajan; no individual proposer field is populated.
• Analytic Number Theory (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/14, pp. 669--746, also listed as "Oberwolfach Report 14/2008": https://doi.org/10.4171/owr/2008/14.
• Supplied reference, "Garaev and Yildirim, zeros of zeta and zeta prime": https://doi.org/10.1093/imrn/rnm091.
• Supplied literature triage, checked 2026-08-21: status "partially solved." It reports quantitative implications by Garaev and Yildirim, calls the stronger Soundararajan equivalence and displayed limiting assertions not fully settled, and describes those implications as involving "prime gaps." The displayed question instead concerns gaps between zeta-zero ordinates; that terminology discrepancy is retained for review. This dated assessment derives from supplied OWR curation; time-sensitive status and cited claims require independent verification.