Expositiones Mathematicae最新文献

{"title":"On the existence of certain Lehmer numbers modulo a prime","authors":"Bidisha Roy","doi":"10.1016/j.exmath.2024.125628","DOIUrl":"10.1016/j.exmath.2024.125628","url":null,"abstract":"<div><div>A <em>Lehmer number modulo an odd prime number</em> <span><math><mi>p</mi></math></span> is a residue class <span><math><mrow><mi>a</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>×</mo></mrow></msubsup></mrow></math></span> whose multiplicative inverse <span><math><mover><mrow><mi>a</mi></mrow><mrow><mo>̄</mo></mrow></mover></math></span> has opposite parity. Lehmer numbers that are also primitive roots are called <em>Lehmer primitive roots</em>. Analogously, in this article we say that a residue class <span><math><mrow><mi>a</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>×</mo></mrow></msubsup></mrow></math></span> is a <em>Lehmer non-primitive root modulo</em> <span><math><mi>p</mi></math></span> if <span><math><mi>a</mi></math></span> is Lehmer number modulo <span><math><mi>p</mi></math></span> which is not a primitive root. We provide explicit estimates for the difference between the number of Lehmer non-primitive roots modulo a prime <span><math><mi>p</mi></math></span> and their “expected number”, which is <span><math><mfrac><mrow><mi>p</mi><mo>−</mo><mn>1</mn><mo>−</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. Similar explicit bounds are also provided for the number of <span><math><mi>k</mi></math></span>-consecutive Lehmer numbers modulo a prime, and <span><math><mi>k</mi></math></span>-consecutive Lehmer primitive roots We also prove that for any prime number <span><math><mrow><mi>p</mi><mo>&gt;</mo><mn>3</mn><mo>.</mo><mn>05</mn><mo>×</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>14</mn></mrow></msup></mrow></math></span>, there exists a Lehmer non-primitive root modulo <span><math><mi>p</mi></math></span>. Moreover, we show that for any positive integer <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span> (respectively, <span><math><mrow><mi>k</mi><mo>≥</mo><mn>5</mn></mrow></math></span>) and for all primes <span><math><mrow><mi>p</mi><mo>≥</mo><mo>exp</mo><mrow><mo>(</mo><mn>1</mn><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> (respectively, <span><math><mrow><mi>p</mi><mo>≥</mo><mo>exp</mo><mrow><mo>(</mo><mn>6</mn><mo>.</mo><mn>8</mn><msup><mrow><mn>7</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>), there exist <span><math><mi>k</mi></math></span> consecutive Lehmer numbers modulo <span><math><mi>p</mi></math></span> (respectively, <span><math><mi>k</mi></math></span> consecutive Lehmer primitive roots modulo <span><math><mi>p</mi></math></span>). For large primes <span><math><mi>p</mi></math></span>, these theorems generalize two results which were proven in a paper by Cohen and Trudgian appeared in the Journal of Number Theory in 2019.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125628"},"PeriodicalIF":0.8,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659468","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":"Strong Gröbner bases and linear algebra in multivariate polynomial rings over Euclidean domains","authors":"Erhard Aichinger","doi":"10.1016/j.exmath.2024.125627","DOIUrl":"10.1016/j.exmath.2024.125627","url":null,"abstract":"<div><div>We provide a self-contained introduction to Gröbner bases of submodules of <span><math><mrow><mi>R</mi><msup><mrow><mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></mrow></mrow><mrow><mi>k</mi></mrow></msup></mrow></math></span>, where <span><math><mi>R</mi></math></span> is a Euclidean domain, and explain how to use these bases to solve linear systems over <span><math><mrow><mi>R</mi><mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125627"},"PeriodicalIF":0.8,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592556","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":"Some remarks on rational right triangles","authors":"Jasbir S. Chahal","doi":"10.1016/j.exmath.2024.125623","DOIUrl":"10.1016/j.exmath.2024.125623","url":null,"abstract":"<div><div>We determine all rational right triangles that tightly enclose the unit circle and the congruent numbers they generate.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125623"},"PeriodicalIF":0.8,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592557","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":"Classification of the conjugacy classes of SL˜(2,R)","authors":"Christian Táfula","doi":"10.1016/j.exmath.2024.125626","DOIUrl":"10.1016/j.exmath.2024.125626","url":null,"abstract":"<div><div>In this note, we classify the conjugacy classes of <span><math><mrow><msub><mrow><mover><mrow><mi>SL</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, the universal covering group of <span><math><mrow><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>. For any non-central element <span><math><mrow><mi>α</mi><mo>∈</mo><msub><mrow><mover><mrow><mi>SL</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, we show that its conjugacy class may be determined by three invariants: (i) <em>Trace</em>: the trace (valued in the set of positive real numbers <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>) of its image <span><math><mover><mrow><mi>α</mi></mrow><mo>¯</mo></mover></math></span> in <span><math><mrow><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>; (ii) <em>Direction type</em>: the sign behavior of the induced self-homeomorphism of <span><math><mi>R</mi></math></span> determined by the lifting <span><math><mrow><msub><mrow><mover><mrow><mi>SL</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>↷</mo><mi>R</mi></mrow></math></span> of the action <span><math><mrow><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>↷</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></math></span>; (iii) <em>The function</em> <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>♯</mi></mrow></msup></math></span>: a conjugacy invariant length function introduced by Mochizuki (2016).</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125626"},"PeriodicalIF":0.8,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659469","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 standard zero-free region for L-functions","authors":"Sun-Kai Leung","doi":"10.1016/j.exmath.2024.125624","DOIUrl":"10.1016/j.exmath.2024.125624","url":null,"abstract":"<div><div>In this short note, we establish a standard zero-free region for a general class of <span><math><mi>L</mi></math></span>-functions for which their logarithms have coefficients with nonnegative real parts, including the Rankin–Selberg <span><math><mi>L</mi></math></span>-functions for unitary cuspidal automorphic representations.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125624"},"PeriodicalIF":0.8,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142525776","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":"Brownian motion in a vector space over a local field is a scaling limit","authors":"Tyler Pierce ,&nbsp;Rahul Rajkumar ,&nbsp;Andrea Stine ,&nbsp;David Weisbart ,&nbsp;Adam M. Yassine","doi":"10.1016/j.exmath.2024.125607","DOIUrl":"10.1016/j.exmath.2024.125607","url":null,"abstract":"<div><div>For any natural number <span><math><mi>d</mi></math></span>, the Vladimirov–Taibleson operator is a natural analogue of the Laplace operator for complex-valued functions on a <span><math><mi>d</mi></math></span>-dimensional vector space <span><math><mi>V</mi></math></span> over a local field <span><math><mi>K</mi></math></span>. Just as the Laplace operator on <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> is the infinitesimal generator of Brownian motion with state space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, the Vladimirov–Taibleson operator on <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></math></span> is the infinitesimal generator of real-time Brownian motion with state space <span><math><mi>V</mi></math></span>. This study deepens the formal analogy between the two types of diffusion processes by demonstrating that both are scaling limits of discrete-time random walks on a discrete group. It generalizes the earlier works, which restricted <span><math><mi>V</mi></math></span> to be the <span><math><mi>p</mi></math></span>-adic numbers.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125607"},"PeriodicalIF":0.8,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432092","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":"Abelian groups acting on the line","authors":"Nancy Guelman,&nbsp;Matilde Martínez","doi":"10.1016/j.exmath.2024.125619","DOIUrl":"10.1016/j.exmath.2024.125619","url":null,"abstract":"<div><div>We study the action of finitely generated Abelian subgroups of <span><math><mrow><mi>H</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mrow><mi>o</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>.</div><div>We propose a presentation where the focus is on understanding the set of stabilizers, which yields a dynamical description of the action that is both elementary and self-contained.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125619"},"PeriodicalIF":0.8,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417662","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":"Some useful tools in the study of nonlinear elliptic problems","authors":"Nikolaos S. Papageorgiou ,&nbsp;Vicenţiu D. Rădulescu","doi":"10.1016/j.exmath.2024.125616","DOIUrl":"10.1016/j.exmath.2024.125616","url":null,"abstract":"<div><p>This paper gives an overview of some basic aspects concerning the qualitative analysis of nonlinear, nonhomogeneous elliptic problems. We are concerned with two classes of elliptic equations with Dirichlet boundary condition. The first problem is driven by a general nonhomogeneous differential operator, which includes several usual operators (such as the <span><math><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></math></span>-Laplace operator introduced by P. Marcellini). Next, we focus on differential operators with unbalanced growth in the nonautonomous case. Our analysis will point out some relevant differences between balanced and unbalanced growth problems. The presentation is done in the context of Dirichlet problems but a similar analysis can be developed for other boundary conditions, such as Neumann or Robin.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125616"},"PeriodicalIF":0.8,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0723086924000835/pdfft?md5=7c5aee29b49c6102b8de52d2790f9ff3&pid=1-s2.0-S0723086924000835-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271042","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":"An introduction to pointwise sparse domination","authors":"Rodrigo Duarte","doi":"10.1016/j.exmath.2024.125605","DOIUrl":"10.1016/j.exmath.2024.125605","url":null,"abstract":"<div><p>The goal of this expository paper is to give a self-contained introduction to sparse domination. This is a method relying on techniques from dyadic Harmonic Analysis which has received a lot of attention in recent years. Essentially, it allows for a unified approach to proving weighted norm inequalities for a large variety of operators. In this work, we will introduce the basic ideas of dyadic Harmonic Analysis, which we use to build up to the main result we discuss on pointwise sparse domination, which is the Lerner–Ombrosi theorem. We also give applications of this theorem to some families of operators, mainly relating to singular integral operators. The text has been structured so as to motivate the introduction of new ideas through the lens of solving specific problems in Harmonic Analysis.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125605"},"PeriodicalIF":0.8,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0723086924000720/pdfft?md5=45dd60e1ad35b7a191fa8e59cc8e5e5d&pid=1-s2.0-S0723086924000720-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271041","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":"Digamma function and general Fischer series in the theory of Kempner sums","authors":"Jean-François Burnol","doi":"10.1016/j.exmath.2024.125604","DOIUrl":"10.1016/j.exmath.2024.125604","url":null,"abstract":"<div><p>The harmonic sum of the integers which are missing <span><math><mi>p</mi></math></span> given digits in a base <span><math><mi>b</mi></math></span> is expressed as <span><math><mrow><mi>b</mi><mo>log</mo><mrow><mo>(</mo><mi>b</mi><mo>)</mo></mrow><mo>/</mo><mi>p</mi></mrow></math></span> plus corrections indexed by the excluded digits and expressed as integrals involving the digamma function and a suitable measure. A number of consequences are derived, such as explicit bounds, monotony, series representations and asymptotic expansions involving the zeta values at integers, and suitable moments of the measure. In the classic Kempner case of <span><math><mrow><mi>b</mi><mo>=</mo><mn>10</mn></mrow></math></span> and 9 as the only excluded digit, the series representation turns out to be exactly identical with a result obtained by Fischer already in 1993. Extending this work is indeed the goal of the present contribution.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125604"},"PeriodicalIF":0.8,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240343","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}