Short mollifiers of the Riemann zeta-function
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arXiv
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| Format: | Preprint |
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2025
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| author | Conrey, J. Brian Farmer, David W. Kwan, Chung-Hang Lin, Yongxiao Turnage-Butterbaugh, Caroline L. |
| author_facet | Conrey, J. Brian Farmer, David W. Kwan, Chung-Hang Lin, Yongxiao Turnage-Butterbaugh, Caroline L. |
| contents | We apply the calculus of variations to construct a new sequence of linear combinations of derivatives of the Riemann $ζ$-function adapted to Levinson's method, which yield a positive proportion of zeros of the $ζ$-function on the critical line, regardless of how short the mollifier is. Our construction extends readily to modular $L$-functions. Even with Levinson's original choice of mollifier, our method more than doubles the proportions of zeros on the critical line for modular $L$-functions previously obtained by Bernard and Kühn--Robles--Zeindler, while relying on the same arithmetic inputs. This indicates that optimizing the linear combinations, an approach that has received relatively little attention, has a more pronounced effect than refining the mollifier when it is short. Curiously, our linear combinations provide non-trivial smooth approximations of Siegel's $\mathfrak{f}$-function in the celebrated Riemann--Siegel formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11108 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Short mollifiers of the Riemann zeta-function Conrey, J. Brian Farmer, David W. Kwan, Chung-Hang Lin, Yongxiao Turnage-Butterbaugh, Caroline L. Number Theory 11M06, 11M26 We apply the calculus of variations to construct a new sequence of linear combinations of derivatives of the Riemann $ζ$-function adapted to Levinson's method, which yield a positive proportion of zeros of the $ζ$-function on the critical line, regardless of how short the mollifier is. Our construction extends readily to modular $L$-functions. Even with Levinson's original choice of mollifier, our method more than doubles the proportions of zeros on the critical line for modular $L$-functions previously obtained by Bernard and Kühn--Robles--Zeindler, while relying on the same arithmetic inputs. This indicates that optimizing the linear combinations, an approach that has received relatively little attention, has a more pronounced effect than refining the mollifier when it is short. Curiously, our linear combinations provide non-trivial smooth approximations of Siegel's $\mathfrak{f}$-function in the celebrated Riemann--Siegel formula. |
| title | Short mollifiers of the Riemann zeta-function |
| topic | Number Theory 11M06, 11M26 |
| url | https://arxiv.org/abs/2508.11108 |