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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2110.06272 |
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| _version_ | 1866909518121664512 |
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| author | Aghili, Farhad |
| author_facet | Aghili, Farhad |
| contents | In this work, we investigate the improper integral of the monomial \(μ(s) = \int_1^{\infty} x^{-s} \,dx \) as a continuous analogue of the infinite series representation of the Riemann $ζ$-function, \(ζ(s) = \sum_{n=1}^{\infty} n^{-s}\). Both the monomial integral and the corresponding series converge for \(\mathrm{Re}(s) > 1\) and diverge for \(s \in \mathbb{C}\) with \(\mathrm{Re}(s) \leq 1\). In this paper, we construct an analytic continuation of the divergent monomial integral to the entire complex plane, excluding a simple pole at \(s = 1\), mirroring the analytic continuation of the $ζ$-function. By performing term-by-term integration of the monomial over successive integer intervals and leveraging Newton's generalization of the binomial theorem, we express the improper integral as a Dirichlet series. This approach establishes an elegant relationship between the \(μ\)-function and the \(ζ\)-function, leading to a functional equation that extends the divergent integral through analytic continuation and that the \(μ\)-function is holomorphic everywhere except at \(s = 1\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_06272 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Analytic Continuation of Divergent Integrals Aghili, Farhad Number Theory In this work, we investigate the improper integral of the monomial \(μ(s) = \int_1^{\infty} x^{-s} \,dx \) as a continuous analogue of the infinite series representation of the Riemann $ζ$-function, \(ζ(s) = \sum_{n=1}^{\infty} n^{-s}\). Both the monomial integral and the corresponding series converge for \(\mathrm{Re}(s) > 1\) and diverge for \(s \in \mathbb{C}\) with \(\mathrm{Re}(s) \leq 1\). In this paper, we construct an analytic continuation of the divergent monomial integral to the entire complex plane, excluding a simple pole at \(s = 1\), mirroring the analytic continuation of the $ζ$-function. By performing term-by-term integration of the monomial over successive integer intervals and leveraging Newton's generalization of the binomial theorem, we express the improper integral as a Dirichlet series. This approach establishes an elegant relationship between the \(μ\)-function and the \(ζ\)-function, leading to a functional equation that extends the divergent integral through analytic continuation and that the \(μ\)-function is holomorphic everywhere except at \(s = 1\). |
| title | Analytic Continuation of Divergent Integrals |
| topic | Number Theory |
| url | https://arxiv.org/abs/2110.06272 |