Journal of Number Theory最新文献

{"title":"A conjecture of Merca on nonnegativity of theta series","authors":"Bing He,&nbsp;Shuming Liu","doi":"10.1016/j.jnt.2024.10.003","DOIUrl":"10.1016/j.jnt.2024.10.003","url":null,"abstract":"<div><div>In this paper, we will study a conjecture of Merca on theta series, which gives a refinement of a conjecture of Andrews and Merca on truncated pentagonal number series. We first show refinements of two special cases of Merca's conjecture and then establish several nonnegativity results on theta series. As applications, we establish positivity results involving two celebrated partition statistics.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 17-36"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bounds for smooth theta sums with rational parameters","authors":"Francesco Cellarosi ,&nbsp;Tariq Osman","doi":"10.1016/j.jnt.2024.10.002","DOIUrl":"10.1016/j.jnt.2024.10.002","url":null,"abstract":"<div><div>We provide explicit families of pairs <span><math><mo>(</mo><mtext>α</mtext><mo>,</mo><mtext>β</mtext><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> such that for sufficiently regular <em>f</em>, there is a constant <em>C</em> for which the theta sum bound<span><span><span><math><mrow><mo>|</mo><munder><mo>∑</mo><mrow><mtext>n</mtext><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></munder><mi>f</mi><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>N</mi></mrow></mfrac><mrow><mi>n</mi></mrow><mo>)</mo></mrow><mi>exp</mi><mo>⁡</mo><mo>{</mo><mn>2</mn><mi>π</mi><mi>i</mi><mo>(</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mo>‖</mo><mrow><mi>n</mi></mrow><mo>‖</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mrow><mi>β</mi></mrow><mo>⋅</mo><mrow><mi>n</mi></mrow><mo>)</mo><mi>x</mi><mo>+</mo><mrow><mi>α</mi></mrow><mo>⋅</mo><mrow><mi>n</mi></mrow><mo>)</mo><mo>}</mo><mo>|</mo></mrow><mspace></mspace><mo>≤</mo><mi>C</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>k</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>,</mo></math></span></span></span> holds for every <span><math><mi>x</mi><mo>∈</mo><mi>R</mi></math></span> and every <span><math><mi>N</mi><mo>∈</mo><mi>N</mi></math></span>. Central to the proof is realising that, for fixed <em>N</em>, the theta sum normalised by <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>k</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span> agrees with an automorphic function <span><math><msub><mrow><mi>Θ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> evaluated along a special curve known as a horocycle lift. The lift depends on the pair <span><math><mo>(</mo><mtext>α</mtext><mo>,</mo><mtext>β</mtext><mo>)</mo></math></span>, and so the bound follows from showing that there are pairs such that <span><math><mo>|</mo><msub><mrow><mi>Θ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>|</mo></math></span> remains bounded along the entire horocycle lift.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 397-426"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Accumulation points of normalized approximations","authors":"Kavita Dhanda,&nbsp;Alan Haynes","doi":"10.1016/j.jnt.2024.09.002","DOIUrl":"10.1016/j.jnt.2024.09.002","url":null,"abstract":"<div><div>Building on classical aspects of the theory of Diophantine approximation, we consider the collection of all accumulation points of normalized integer vector translates of points <span><math><mi>q</mi><mi>α</mi></math></span> with <span><math><mi>α</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> and <span><math><mi>q</mi><mo>∈</mo><mi>Z</mi></math></span>. In the first part of the paper we derive measure theoretic and Hausdorff dimension results about the set of <strong><em>α</em></strong> whose accumulation points are all of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. In the second part we focus primarily on the case when the coordinates of <strong><em>α</em></strong> together with 1 form a basis for an algebraic number field <em>K</em>. Here we show that, under the correct normalization, the set of accumulation points displays an ordered geometric structure which reflects algebraic properties of the underlying number field. For example, when <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>, this collection of accumulation points can be described as a countable union of dilates (by norms of elements of an order in <em>K</em>) of a single ellipse, or of a pair of hyperbolas, depending on whether or not <em>K</em> has a non-trivial embedding into <span><math><mi>C</mi></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 1-38"},"PeriodicalIF":0.6,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Diophantine equation 2s + pk = m2 with a Fermat prime p","authors":"Florian Luca ,&nbsp;István Pink","doi":"10.1016/j.jnt.2024.09.006","DOIUrl":"10.1016/j.jnt.2024.09.006","url":null,"abstract":"<div><div>In this paper, we find all the solutions of the Diophantine equation from the title.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 49-71"},"PeriodicalIF":0.6,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Statistics for 3-isogeny induced Selmer groups of elliptic curves","authors":"Pratiksha Shingavekar","doi":"10.1016/j.jnt.2024.09.003","DOIUrl":"10.1016/j.jnt.2024.09.003","url":null,"abstract":"<div><div>Given a sixth power free integer <em>a</em>, let <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> be the elliptic curve defined by <span><math><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>a</mi></math></span>. We prove explicit results for the lower density of sixth power free integers <em>a</em> for which the 3-isogeny induced Selmer group of <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> over <span><math><mi>Q</mi><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span> has dimension ≤1. The results are proven by refining the strategy of Davenport–Heilbronn, by relating the statistics for integral binary cubic forms to the statistics for 3-isogeny induced Selmer groups.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 72-94"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Construction and non-vanishing of a family of vector-valued Siegel Poincaré series","authors":"Sonja Žunar","doi":"10.1016/j.jnt.2024.09.007","DOIUrl":"10.1016/j.jnt.2024.09.007","url":null,"abstract":"<div><div>Using Poincaré series of <em>K</em>-finite matrix coefficients of integrable antiholomorphic discrete series representations of <span><math><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, we construct a spanning set for the space <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> of Siegel cusp forms of weight <em>ρ</em> for Γ, where <em>ρ</em> is an irreducible polynomial representation of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> of highest weight <span><math><mi>ω</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mo>…</mo><mo>≥</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>&gt;</mo><mn>2</mn><mi>n</mi></math></span>, and Γ is a discrete subgroup of <span><math><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> commensurable with <span><math><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. Moreover, using a variant of Muić's integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel Poincaré series.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 95-123"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On eventually greedy best underapproximations by Egyptian fractions","authors":"Vjekoslav Kovač","doi":"10.1016/j.jnt.2024.09.004","DOIUrl":"10.1016/j.jnt.2024.09.004","url":null,"abstract":"<div><div>Erdős and Graham found it conceivable that the best <em>n</em>-term Egyptian underapproximation of almost every positive number for sufficiently large <em>n</em> gets constructed in a greedy manner, i.e., from the best <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-term Egyptian underapproximation. We show that the opposite is true: the set of real numbers with this property has Lebesgue measure zero.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 39-48"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Improved bounds for the index conjecture in zero-sum theory","authors":"Andrew Pendleton","doi":"10.1016/j.jnt.2024.09.005","DOIUrl":"10.1016/j.jnt.2024.09.005","url":null,"abstract":"<div><div>The Index Conjecture in zero-sum theory states that when <em>n</em> is coprime to 6 and <em>k</em> equals 4, every minimal zero-sum sequence of length <em>k</em> modulo <em>n</em> has index 1. While other values of <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> have been studied thoroughly in the last 30 years, it is only recently that the conjecture has been proven for <span><math><mi>n</mi><mo>&gt;</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>20</mn></mrow></msup></math></span>. In this paper, we prove that said upper bound can be reduced to <span><math><mn>4.6</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>13</mn></mrow></msup></math></span>, and lower under certain coprimality conditions. Further, we verify the conjecture for <span><math><mi>n</mi><mo>&lt;</mo><mn>1.8</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>6</mn></mrow></msup></math></span> through the application of High Performance Computing (HPC).</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 124-141"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Common values of linear recurrences related to Shank's simplest cubics","authors":"Attila Pethő ,&nbsp;Szabolcs Tengely","doi":"10.1016/j.jnt.2024.09.001","DOIUrl":"10.1016/j.jnt.2024.09.001","url":null,"abstract":"<div><div>Let <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><mi>Z</mi></math></span> not all zeroes and let <span><math><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>F</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> be the linear recursive sequence, which is defined by the initial terms <span><math><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>A</mi><mo>,</mo><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>B</mi><mo>,</mo><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mn>2</mn><mo>)</mo><mo>=</mo><mi>C</mi></math></span> and whose characteristic polynomial is Daniel Shanks simplest cubic <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mo>(</mo><mi>u</mi><mo>−</mo><mn>1</mn><mo>)</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mo>(</mo><mi>u</mi><mo>+</mo><mn>2</mn><mo>)</mo><mi>X</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>u</mi><mo>∈</mo><mi>Z</mi></math></span>. We prove that there exists an effectively computable constant <em>c</em> depending only on <span><math><mi>L</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mo>|</mo><mi>A</mi><mo>|</mo><mo>,</mo><mo>|</mo><mi>B</mi><mo>|</mo><mo>,</mo><mo>|</mo><mi>C</mi><mo>|</mo><mo>}</mo></math></span> such that if <span><math><mo>|</mo><mi>F</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>|</mo><mo>=</mo><mo>|</mo><mi>F</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>|</mo></math></span> holds for some integers <span><math><mi>u</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>m</mi></math></span> with <span><math><mi>n</mi><mo>≠</mo><mi>m</mi></math></span> then <span><math><mo>|</mo><mi>n</mi><mo>|</mo><mo>,</mo><mo>|</mo><mi>m</mi><mo>|</mo><mo>&lt;</mo><mi>c</mi></math></span>. For the choices <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo><mo>∈</mo><mo>{</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>}</mo></math></span> we solve the above equations completely. At the end we give an outlook to the equation <span><math><mi>F</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>F</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>v</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> for some fixed integers <span><math><mi>n</mi><mo>,</mo><mi>m</mi></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"267 ","pages":"Pages 34-79"},"PeriodicalIF":0.6,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the number of prime factors with a given multiplicity over h-free and h-full numbers","authors":"Sourabhashis Das,&nbsp;Wentang Kuo,&nbsp;Yu-Ru Liu","doi":"10.1016/j.jnt.2024.08.007","DOIUrl":"10.1016/j.jnt.2024.08.007","url":null,"abstract":"<div><div>Let <em>k</em> and <em>n</em> be natural numbers. Let <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of distinct prime factors of <em>n</em> with multiplicity <em>k</em> as studied by Elma and the third author <span><span>[5]</span></span>. We obtain asymptotic estimates for the first and the second moments of <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> when restricted to the set of <em>h</em>-free and <em>h</em>-full numbers. We prove that <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> has normal order <span><math><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi></math></span> over <em>h</em>-free numbers, <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> has normal order <span><math><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi></math></span> over <em>h</em>-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> with <span><math><mn>1</mn><mo>&lt;</mo><mi>k</mi><mo>&lt;</mo><mi>h</mi></math></span> do not have normal order over <em>h</em>-free numbers and <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> with <span><math><mi>k</mi><mo>&gt;</mo><mi>h</mi></math></span> do not have normal order over <em>h</em>-full numbers.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"267 ","pages":"Pages 176-201"},"PeriodicalIF":0.6,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328287","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}