Large values of $L(σ,χ)$ for subgroups of characters
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915912899100672 |
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| author | Darbar, Pranendu Kerr, Bryce Munsch, Marc Shparlinski, Igor |
| author_facet | Darbar, Pranendu Kerr, Bryce Munsch, Marc Shparlinski, Igor |
| contents | We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,χ)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $χ$ runs through a thin subgroup of all characters modulo an integer $q$. Some of these bounds are based on new zero-density estimates on average over a subgroup of characters. These bounds follow from a mean value estimate for character sums, which is based on the work of D. R. Heath-Brown (1979). As yet another application of this mean value estimate, we obtain an unconditional version of a conditional (on the Generalised Riemann Hypothesis) result of Z. Rudnick and A. Zaharescu (2000) about gaps between primitive roots. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02960 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Large values of $L(σ,χ)$ for subgroups of characters Darbar, Pranendu Kerr, Bryce Munsch, Marc Shparlinski, Igor Number Theory We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,χ)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $χ$ runs through a thin subgroup of all characters modulo an integer $q$. Some of these bounds are based on new zero-density estimates on average over a subgroup of characters. These bounds follow from a mean value estimate for character sums, which is based on the work of D. R. Heath-Brown (1979). As yet another application of this mean value estimate, we obtain an unconditional version of a conditional (on the Generalised Riemann Hypothesis) result of Z. Rudnick and A. Zaharescu (2000) about gaps between primitive roots. |
| title | Large values of $L(σ,χ)$ for subgroups of characters |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.02960 |