Integral Means Spectrum for the Random Riemann Zeta Function
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910078713462784 |
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| author | Duplantier, Bertrand Gayrard, Véronique Saksman, Eero |
| author_facet | Duplantier, Bertrand Gayrard, Véronique Saksman, Eero |
| contents | We study the integral means spectrum associated with the analytic function whose derivative is the so-called randomized Riemann zeta-function, introduced some time ago by Bagchi. The randomized $ζ$-function, $ζ_{\mathrm{rand}}(σ+ih)$, is known to represent the asymptotic statistical behaviour of the random vertical shifts of the actual $ζ$-function in the critical strip, $1/2 <σ\leq 1, h\in \mathbb R$, and appears in a number of recent works on the asymptotic behavior of the moments and maxima of the $ζ$-function on short intervals along the critical axis $σ=1/2$. Using probability and basic analytic number theory, we show that the complex integral means spectrum of the primitive of $ζ_{\mathrm{rand}}$ is almost surely of the form conjectured 30 years ago by Kraetzer, for the so-called universal integral means spectrum of univalent functions in the disc. The Riemann $ζ$-function and its random version have recently been rigorously related to the so-called Gaussian multiplicative chaos (GMC), initiated by Kahane 40 years ago. In the case of the holomorphic multiplicative chaos on the unit disc -- an important stochastic object closely related to Liouville quantum gravity on the unit circle -- we prove that the integral means spectrum of the primitive is almost surely also of the same Kraetzer form. However, we establish that neither the primitive of the random function $ζ_{\mathrm{rand}}$, nor that of the holomorphic GMC are injective. Building on earlier work by one of the authors and Webb on the convergence of Riemann $ζ$-function on the critical line to a holomorphic GMC distribution, we finally provide an alternative derivation of the integral means spectrum for the random Riemann $ζ$-function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_26507 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Integral Means Spectrum for the Random Riemann Zeta Function Duplantier, Bertrand Gayrard, Véronique Saksman, Eero Complex Variables Mathematical Physics Number Theory Probability 60G15, 60G60, 60G70, 11M06 We study the integral means spectrum associated with the analytic function whose derivative is the so-called randomized Riemann zeta-function, introduced some time ago by Bagchi. The randomized $ζ$-function, $ζ_{\mathrm{rand}}(σ+ih)$, is known to represent the asymptotic statistical behaviour of the random vertical shifts of the actual $ζ$-function in the critical strip, $1/2 <σ\leq 1, h\in \mathbb R$, and appears in a number of recent works on the asymptotic behavior of the moments and maxima of the $ζ$-function on short intervals along the critical axis $σ=1/2$. Using probability and basic analytic number theory, we show that the complex integral means spectrum of the primitive of $ζ_{\mathrm{rand}}$ is almost surely of the form conjectured 30 years ago by Kraetzer, for the so-called universal integral means spectrum of univalent functions in the disc. The Riemann $ζ$-function and its random version have recently been rigorously related to the so-called Gaussian multiplicative chaos (GMC), initiated by Kahane 40 years ago. In the case of the holomorphic multiplicative chaos on the unit disc -- an important stochastic object closely related to Liouville quantum gravity on the unit circle -- we prove that the integral means spectrum of the primitive is almost surely also of the same Kraetzer form. However, we establish that neither the primitive of the random function $ζ_{\mathrm{rand}}$, nor that of the holomorphic GMC are injective. Building on earlier work by one of the authors and Webb on the convergence of Riemann $ζ$-function on the critical line to a holomorphic GMC distribution, we finally provide an alternative derivation of the integral means spectrum for the random Riemann $ζ$-function. |
| title | Integral Means Spectrum for the Random Riemann Zeta Function |
| topic | Complex Variables Mathematical Physics Number Theory Probability 60G15, 60G60, 60G70, 11M06 |
| url | https://arxiv.org/abs/2603.26507 |