For a prime number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_761_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> we let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_761_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">$G(p)=\left\langle \overline{x}\right\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfenced close="〉" open="〈"> <mover> <mi>x</mi> <mo>¯</mo> </mover> </mfenced> </mrow> </math></EquationSource> </InlineEquation> denote the group of reduced residue classes modulo p and we let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_761_Article_IEq3.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="189" /> </InlineMediaObject> <EquationSource Format="TEX">$\widehat{G}(p)=\left\{ \chi _{0},\chi _{1},\ldots ,\chi _{p-2}\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>G</mi> <mo stretchy="true">^</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfenced close="}" open="{"> <msub> <mi>χ</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>χ</mi> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> </mfenced> </mrow> </math></EquationSource> </InlineEquation> denote the group of Dirichlet characters modulo p. Let l and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_761_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu _l$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ν</mi> <mi>l</mi> </msub> </math></EquationSource> </InlineEquation> be integers such that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_761_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">$\gcd (p,l)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_761_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$\overline{l}=\overline{x}^{\nu _l}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>l</mi> <mo>¯</mo> </mover> <mo>=</mo> <msup> <mover> <mi>x</mi> <mo>¯</mo> </mover> <msub> <mi>ν</mi> <mi>l</mi> </msub> </msup> </mrow> </math></EquationSource> </InlineEquation>. The main purpose of this paper is to present an explicit formula for the sum: <Equation ID="Equ12"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_761_Article_Equ12.gif" Format="GIF" Height="52" Rendition="HTML" Resolution="72" Type="Linedraw" Width="196" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \sum _{a=0}^{p-2}|B_1(\chi _a)|^2\cos \left( \dfrac{2\pi a\nu _l}{p-1}\right) , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </munderover> <msup> <mrow> <mo stretchy="false">|</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>χ</mi> <mi>a</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mo>cos</mo> <mfenced close=")" open="("> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> <msub> <mi>ν</mi> <mi>l</mi> </msub> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_761_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$B_m(\chi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>χ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_761_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$(m\ge 0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>≥</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are the generalized Bernoulli numbers associated with <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_761_Article_IEq9.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\chi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation>.

A new identity for cosine function and generalized Bernoulli numbers

期刊论文
发表日期：

2025年2月10日

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Brahim Mittou

摘要

For a prime number $p\ge 3$ $p \geq 3$ we let $G(p)=\left\langle \overline{x}\right\rangle $ $G (p) = 〈\bar{x}〉$ denote the group of reduced residue classes modulo p and we let $\widehat{G}(p)=\left\{ \chi _{0},\chi _{1},\ldots ,\chi _{p-2}\right\} $ $\hat{G} (p) = \{χ_{0}, χ_{1}, \dots, χ_{p - 2}\}$ denote the group of Dirichlet characters modulo p. Let l and $\nu _l$ $ν_{l}$ be integers such that $\gcd (p,l)=1$ $gcd (p, l) = 1$ and $\overline{l}=\overline{x}^{\nu _l}$ $\bar{l} = {\bar{x}}^{ν_{l}}$ . The main purpose of this paper is to present an explicit formula for the sum: $\begin{aligned} \sum _{a=0}^{p-2}|B_1(\chi _a)|^2\cos \left( \dfrac{2\pi a\nu _l}{p-1}\right) , \end{aligned}$ $\begin{matrix} \sum_{a = 0}^{p - 2} {| B_{1} (χ_{a}) |}^{2} cos (\frac{2 π a ν_{l}}{p - 1}), \end{matrix}$ where $B_m(\chi )$ $B_{m} (χ)$ $(m\ge 0)$ $(m \geq 0)$ are the generalized Bernoulli numbers associated with $\chi $ $χ$ .