Riemann's auxiliary Function. Basic Results
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909216706396160 |
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| author | de Reyna, J. Arias |
| author_facet | de Reyna, J. Arias |
| contents | We give the definition, main properties and integral expressions of the auxiliary function of Riemann $\mathop{\mathcal R }(s)$. For example we prove $$π^{-s/2}Γ(s/2)\mathop{\mathcal R }(s)=-\frac{e^{-πi s/4}}{ s}\int_{-1}^{-1+i\infty} τ^{s/2}\vartheta_3'(τ)\,dτ.$$ Many of these results are known, but they serve as a reference. We give the values of $\mathop{\mathcal R }(s)$ at integers except at odd natural numbers. We have $$ζ(\tfrac12+it)=e^{-i\vartheta(t)}Z(t),\quad \mathop{\mathcal R }(\tfrac12+it)=\tfrac12e^{-i\vartheta(t)}(Z(t)+iY(t)),$$ with $\vartheta(t)$, $Z(t)$ and $Y(t)$ real functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02403 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Riemann's auxiliary Function. Basic Results de Reyna, J. Arias History and Overview Number Theory Primary 11M06, Secondary 30D10 We give the definition, main properties and integral expressions of the auxiliary function of Riemann $\mathop{\mathcal R }(s)$. For example we prove $$π^{-s/2}Γ(s/2)\mathop{\mathcal R }(s)=-\frac{e^{-πi s/4}}{ s}\int_{-1}^{-1+i\infty} τ^{s/2}\vartheta_3'(τ)\,dτ.$$ Many of these results are known, but they serve as a reference. We give the values of $\mathop{\mathcal R }(s)$ at integers except at odd natural numbers. We have $$ζ(\tfrac12+it)=e^{-i\vartheta(t)}Z(t),\quad \mathop{\mathcal R }(\tfrac12+it)=\tfrac12e^{-i\vartheta(t)}(Z(t)+iY(t)),$$ with $\vartheta(t)$, $Z(t)$ and $Y(t)$ real functions. |
| title | Riemann's auxiliary Function. Basic Results |
| topic | History and Overview Number Theory Primary 11M06, Secondary 30D10 |
| url | https://arxiv.org/abs/2406.02403 |