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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.04038 |
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| _version_ | 1866917713441456128 |
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| author | de Reyna, Juan Arias |
| author_facet | de Reyna, Juan Arias |
| contents | Starting from some of Norman Levinson's results, we construct interesting examples of functions $f(s)$ such that for $s=\frac12+it$, we have $Z(t)=2\Re\{π^{-\frac{s}{2}}Γ(s/2)f(s)\}$. For example one such function is \[\begin{aligned}{\mathcal R }_{-3}(s)=\frac12&\int_{0\swarrow1}\frac{x^{-s}e^{3πix^2}}{e^{πi x}-e^{-πi x}}\,dx\\&+\frac{1}{2\sqrt{3}}\int_{0\swarrow1}\frac{x^{-s}e^{\frac{πi}{3}x^2}}{e^{πi x}-e^{-πi x}}\Bigl(e^{\frac{πi}{2}}+2e^{-\frac{πi}{6}}\cos(\tfrac{2πx}{3})\Bigr)\,dx.\end{aligned}\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04038 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Levinson Functions de Reyna, Juan Arias Number Theory Primary 11M06, Secondary 30D99 Starting from some of Norman Levinson's results, we construct interesting examples of functions $f(s)$ such that for $s=\frac12+it$, we have $Z(t)=2\Re\{π^{-\frac{s}{2}}Γ(s/2)f(s)\}$. For example one such function is \[\begin{aligned}{\mathcal R }_{-3}(s)=\frac12&\int_{0\swarrow1}\frac{x^{-s}e^{3πix^2}}{e^{πi x}-e^{-πi x}}\,dx\\&+\frac{1}{2\sqrt{3}}\int_{0\swarrow1}\frac{x^{-s}e^{\frac{πi}{3}x^2}}{e^{πi x}-e^{-πi x}}\Bigl(e^{\frac{πi}{2}}+2e^{-\frac{πi}{6}}\cos(\tfrac{2πx}{3})\Bigr)\,dx.\end{aligned}\] |
| title | Levinson Functions |
| topic | Number Theory Primary 11M06, Secondary 30D99 |
| url | https://arxiv.org/abs/2407.04038 |