Functional equation for LC-functions with even or odd modulator
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914205969416192 |
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| author | Lamgouni, Lahcen |
| author_facet | Lamgouni, Lahcen |
| contents | In a recent work, we introduced \textit{LC-functions} $L(s,f)$, associated to a certain real-analytic function $f$ at $0$, extending the concept of the Hurwitz zeta function and its formula. In this paper, we establish the existence of a functional equation for a specific class of LC-functions. More precisely, we demonstrate that if the function $p_f(t):=f(t)(e^t-1)/t$, called the \textit{modulator} of $L(s,f)$, exhibits even or odd symmetry, the \textit{LC-function formula} -- a generalization of the Hurwitz formula -- naturally simplifies to a functional equation analogous to that of the Dirichlet L-function $L(s,χ)$, associated to a primitive character $χ$ of inherent parity. Furthermore, using this equation, we derive a general formula for the values of these LC-functions at even or odd positive integers, depending on whether the modulator $p_f$ is even or odd, respectively. Two illustrative examples of the functional equation are provided for distinct parity of modulators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_00813 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Functional equation for LC-functions with even or odd modulator Lamgouni, Lahcen Number Theory Complex Variables 11M41, 11M06, 11M35 In a recent work, we introduced \textit{LC-functions} $L(s,f)$, associated to a certain real-analytic function $f$ at $0$, extending the concept of the Hurwitz zeta function and its formula. In this paper, we establish the existence of a functional equation for a specific class of LC-functions. More precisely, we demonstrate that if the function $p_f(t):=f(t)(e^t-1)/t$, called the \textit{modulator} of $L(s,f)$, exhibits even or odd symmetry, the \textit{LC-function formula} -- a generalization of the Hurwitz formula -- naturally simplifies to a functional equation analogous to that of the Dirichlet L-function $L(s,χ)$, associated to a primitive character $χ$ of inherent parity. Furthermore, using this equation, we derive a general formula for the values of these LC-functions at even or odd positive integers, depending on whether the modulator $p_f$ is even or odd, respectively. Two illustrative examples of the functional equation are provided for distinct parity of modulators. |
| title | Functional equation for LC-functions with even or odd modulator |
| topic | Number Theory Complex Variables 11M41, 11M06, 11M35 |
| url | https://arxiv.org/abs/2409.00813 |