Joint extreme values of $L$-functions on and off the critical line
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917462154412032 |
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| author | Sourmelidis, Athanasios |
| author_facet | Sourmelidis, Athanasios |
| contents | It is shown that any number of distinct primitive $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$ $L$-functions can simultaneously attain large values on the critical line. This is an unconditional improvement of a general result due to Heap and Li who have assumed the Riemann Hypothesis for more than three such $L$-functions. The joint distribution of $\mathrm{GL}(m)$ $L$-functions to the right of the critical line is also studied under certain zero-density estimates. In particular, we can partially recover results of Inoue and Li on Dirichlet $L$-functions and generally improve upon the work of Mahatab, Pańkowski and Vatwani on the class of $L$-functions introduced by Selberg. The main machinery in both cases, on and off the critical line, is the resonance method of Soundararajan and Hilberdink/Voronin, respectively. On the critical line we additionally introduce a variation of Heath-Brown's method for the fractional moments of the Riemann zeta-function which makes it possible to avoid using any information on the zero distribution of $L$-functions whose degree is less than three. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_03665 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Joint extreme values of $L$-functions on and off the critical line Sourmelidis, Athanasios Number Theory 11M06, 11F66 It is shown that any number of distinct primitive $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$ $L$-functions can simultaneously attain large values on the critical line. This is an unconditional improvement of a general result due to Heap and Li who have assumed the Riemann Hypothesis for more than three such $L$-functions. The joint distribution of $\mathrm{GL}(m)$ $L$-functions to the right of the critical line is also studied under certain zero-density estimates. In particular, we can partially recover results of Inoue and Li on Dirichlet $L$-functions and generally improve upon the work of Mahatab, Pańkowski and Vatwani on the class of $L$-functions introduced by Selberg. The main machinery in both cases, on and off the critical line, is the resonance method of Soundararajan and Hilberdink/Voronin, respectively. On the critical line we additionally introduce a variation of Heath-Brown's method for the fractional moments of the Riemann zeta-function which makes it possible to avoid using any information on the zero distribution of $L$-functions whose degree is less than three. |
| title | Joint extreme values of $L$-functions on and off the critical line |
| topic | Number Theory 11M06, 11F66 |
| url | https://arxiv.org/abs/2605.03665 |