Indagationes Mathematicae-New Series最新文献

{"title":"Box dimension of generic Hölder level sets","authors":"Zoltán Buczolich ,&nbsp;Balázs Maga","doi":"10.1016/j.indag.2024.03.015","DOIUrl":"10.1016/j.indag.2024.03.015","url":null,"abstract":"<div><p>Hausdorff dimension of level sets of generic continuous functions defined on fractals can give information about the “thickness/narrow cross-sections” of a “network” corresponding to a fractal set. This leads to the definition of the topological Hausdorff dimension of fractals. Finer information might be obtained by considering the Hausdorff dimension of level sets of generic 1-Hölder-<span><math><mi>α</mi></math></span> functions, which has a stronger dependence on the geometry of the fractal, as displayed in our previous papers (Buczolich et al., 2022 [9,10]). In this paper, we extend our investigations to the lower and upper box-counting dimensions as well: while the former yields results highly resembling the ones about the Hausdorff dimension of level sets, the latter exhibits a different behavior. Instead of “finding narrow-cross sections”, results related to upper box-counting dimension “measure” how much level sets can spread out on the fractal, and how widely the generic function can “oscillate” on it. Key differences are illustrated by giving estimates concerning the Sierpiński triangle.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 531-554"},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000326/pdfft?md5=5a77b60c431ef034f802646912c24066&pid=1-s2.0-S0019357724000326-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Root numbers of a family of elliptic curves and two applications","authors":"Jonathan Love","doi":"10.1016/j.indag.2024.04.003","DOIUrl":"10.1016/j.indag.2024.04.003","url":null,"abstract":"<div><p>For each <span><math><mrow><mi>t</mi><mo>∈</mo><mi>Q</mi><mo>∖</mo><mrow><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>, define an elliptic curve over <span><math><mi>Q</mi></math></span> by <span><span><span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>x</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>.</mo></mrow></math></span></span></span>Using a formula for the root number <span><math><mrow><mi>W</mi><mrow><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> as a function of <span><math><mi>t</mi></math></span> and assuming some standard conjectures about ranks of elliptic curves, we determine (up to a set of density zero) the set of isomorphism classes of elliptic curves <span><math><mrow><mi>E</mi><mo>/</mo><mi>Q</mi></mrow></math></span> whose Mordell–Weil group contains <span><math><mrow><mi>Z</mi><mo>×</mo><mi>Z</mi><mo>/</mo><mn>2</mn><mi>Z</mi><mo>×</mo><mi>Z</mi><mo>/</mo><mn>4</mn><mi>Z</mi></mrow></math></span>, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 555-569"},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000351/pdfft?md5=2bd90ba3afb1d531934bbb073c1710e2&pid=1-s2.0-S0019357724000351-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Each friend of 10 has at least 10 nonidentical prime factors","authors":"Henry (Maya) Robert Thackeray","doi":"10.1016/j.indag.2024.04.011","DOIUrl":"10.1016/j.indag.2024.04.011","url":null,"abstract":"<div><p>For each positive integer <span><math><mi>n</mi></math></span>, if the sum of the factors of <span><math><mi>n</mi></math></span> is divided by <span><math><mi>n</mi></math></span>, then the result is called the abundancy index of <span><math><mi>n</mi></math></span>. If the abundancy index of some positive integer <span><math><mi>m</mi></math></span> equals the abundancy index of <span><math><mi>n</mi></math></span> but <span><math><mi>m</mi></math></span> is not equal to <span><math><mi>n</mi></math></span>, then <span><math><mi>m</mi></math></span> and <span><math><mi>n</mi></math></span> are called friends. A positive integer with no friends is called solitary. The smallest positive integer that is not known to have a friend and is not known to be solitary is 10.</p><p>It is not known if the number 6 has odd friends, that is, if odd perfect numbers exist. In a 2007 article, Nielsen proved that the number of nonidentical prime factors in any odd perfect number is at least 9. A 2015 article by Nielsen, which was more complicated and used a computer program that took months to complete, increased the lower bound from 9 to 10.</p><p>This work applies methods from Nielsen’s 2007 article to show that each friend of 10 has at least 10 nonidentical prime factors.</p><p>This is a formal write-up of results presented at the Southern Africa Mathematical Sciences Association Conference 2023 at the University of Pretoria.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 595-607"},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000430/pdfft?md5=5f3ad739533e1db88fb550301881c997&pid=1-s2.0-S0019357724000430-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantum superintegrable spin systems on graph connections","authors":"Nicolai Reshetikhin ,&nbsp;Jasper Stokman","doi":"10.1016/j.indag.2024.03.008","DOIUrl":"10.1016/j.indag.2024.03.008","url":null,"abstract":"<div><div>In this paper we construct certain quantum spin systems on moduli spaces of <span><math><mi>G</mi></math></span>-connections on a connected oriented finite graph, with <span><math><mi>G</mi></math></span> a simply connected compact Lie group. We construct joint eigenfunctions of the commuting quantum Hamiltonians in terms of local invariant tensors. We determine sufficient conditions ensuring superintegrability of the quantum spin system using irreducibility criteria for Harish-Chandra modules due to Harish-Chandra and Lepowsky &amp; McCollum. The resulting class of quantum superintegrable spin systems includes the quantum periodic and open spin Calogero–Moser spin chains as special cases. In the periodic case the description of the joint eigenfunctions in terms of local invariant tensors are multipoint generalized trace functions, in the open case multipoint spherical functions on compact symmetric spaces.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 2","pages":"Pages 644-674"},"PeriodicalIF":0.5,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Symplectic complexity of reductive group actions","authors":"Avraham Aizenbud,&nbsp;Dmitry Gourevitch","doi":"10.1016/j.indag.2024.03.010","DOIUrl":"10.1016/j.indag.2024.03.010","url":null,"abstract":"<div><div>Let a complex algebraic reductive group <span><math><mi>G</mi></math></span> act on a complex algebraic manifold <span><math><mi>X</mi></math></span>. For a <span><math><mi>G</mi></math></span>-invariant subvariety <span><math><mi>Ξ</mi></math></span> of the nilpotent cone <span><math><mrow><mi>N</mi><mrow><mo>(</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>∗</mo></mrow></msup><mo>)</mo></mrow><mo>⊂</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span> we define a notion of <span><math><mi>Ξ</mi></math></span>-symplectic complexity of <span><math><mi>X</mi></math></span>. This notion generalizes the notion of complexity defined in Vinberg (1986). We prove several properties of this notion, and relate it to the notion of <span><math><mi>Ξ</mi></math></span>-complexity defined in Aizenbud and Gourevitch (2024) motivated by its relation with representation theory.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 2","pages":"Pages 703-712"},"PeriodicalIF":0.5,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generating operators of symmetry breaking—From discrete to continuous","authors":"Toshiyuki Kobayashi","doi":"10.1016/j.indag.2024.03.007","DOIUrl":"10.1016/j.indag.2024.03.007","url":null,"abstract":"<div><div>Based on the “generating operator” of the Rankin–Cohen brackets introduced in Kobayashi–Pevzner [arXiv:2306.16800], we present a method to construct various fundamental operators with continuous parameters such as invariant trilinear forms on infinite-dimensional representations, the Fourier and the Poisson transforms on the anti-de Sitter space, and integral symmetry breaking<span><span> operators for the fusion rules, among others, out of a countable set of differential </span>symmetry breaking operators.</span></div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 2","pages":"Pages 631-643"},"PeriodicalIF":0.5,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Symmetric pairs and branching laws","authors":"Paul-Émile Paradan","doi":"10.1016/j.indag.2024.03.009","DOIUrl":"10.1016/j.indag.2024.03.009","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span><span> be a compact connected Lie group and let </span><span><math><mi>H</mi></math></span> be a subgroup fixed by an involution. A classical result assures that the <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>ℂ</mi></mrow></msub></math></span>-action on the flag variety <span><math><mi>F</mi></math></span> of <span><math><mi>G</mi></math></span> admits a finite number of orbits. In this article we propose a formula for the branching coefficients of the symmetric pair <span><math><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></math></span> that is parametrized by <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>ℂ</mi></mrow></msub><mo>∖</mo><mi>F</mi></mrow></math></span>.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 2","pages":"Pages 675-702"},"PeriodicalIF":0.5,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the separation of the roots of the generalized Fibonacci polynomial","authors":"Jonathan García ,&nbsp;Carlos A. Gómez ,&nbsp;Florian Luca","doi":"10.1016/j.indag.2023.12.002","DOIUrl":"10.1016/j.indag.2023.12.002","url":null,"abstract":"<div><p>In this paper we prove some separation results for the roots of the generalized Fibonacci polynomials and their absolute values.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 2","pages":"Pages 269-281"},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138679718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on the Diophantine equations 2ln2=1+q+⋯+qα and application to odd perfect numbers","authors":"Yoshinosuke Hirakawa","doi":"10.1016/j.indag.2023.12.004","DOIUrl":"10.1016/j.indag.2023.12.004","url":null,"abstract":"<div><p>Let <span><math><mi>N</mi></math></span> be an odd perfect number. Then, Euler proved that there exist some integers <span><math><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></math></span> and a prime <span><math><mi>q</mi></math></span> such that <span><math><mrow><mi>N</mi><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>q</mi><mo>∤</mo><mi>n</mi></mrow></math></span>, and <span><math><mrow><mi>q</mi><mo>≡</mo><mi>α</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>mod</mo><mspace></mspace><mn>4</mn></mrow></math></span>. In this note, we prove that the ratio <span><math><mfrac><mrow><mi>σ</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></math></span> is neither a square nor a square times a single prime unless <span><math><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow></math></span><span>. It is a direct consequence of a certain property of the Diophantine equation </span><span><math><mrow><mn>2</mn><mi>l</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>+</mo><mi>q</mi><mo>+</mo><mo>⋯</mo><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></math></span>, where <span><math><mi>l</mi></math></span><span> denotes one or a prime, and its proof is based on the prime ideal factorization in the quadratic orders </span><span><math><mrow><mi>Z</mi><mrow><mo>[</mo><msqrt><mrow><mn>1</mn><mo>−</mo><mi>q</mi></mrow></msqrt><mo>]</mo></mrow></mrow></math></span> and the primitive solutions of generalized Fermat equations <span><math><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>=</mo><mn>2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>. We give also a slight generalization to odd multiply perfect numbers.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 2","pages":"Pages 282-287"},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139104891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On some coefficients of the Artin–Hasse series modulo a prime","authors":"Marina Avitabile,&nbsp;Sandro Mattarei","doi":"10.1016/j.indag.2024.01.003","DOIUrl":"10.1016/j.indag.2024.01.003","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be an odd prime, and let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be the reduction modulo &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of the Artin–Hasse exponential series. We obtain a polynomial expression for &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; in terms of those &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, for even &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. A conjectural analogue covering the case of odd &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; can be stated in various polynomial forms, essentially in terms of the polynomial &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; denotes the &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;th Bernoulli number.&lt;/p&gt;&lt;p&gt;We prove that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; satisfies the functional equation &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;£&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;£&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are the truncated logarithm and the Wilson quotient. This is an analogue modulo &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of a functional equation, in &lt;span&gt;","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 2","pages":"Pages 317-328"},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139506915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}