Journal of Number Theory最新文献

{"title":"Extended modular functions and definite form class groups","authors":"Ho Yun Jung ,&nbsp;Ja Kyung Koo ,&nbsp;Dong Hwa Shin ,&nbsp;Gyucheol Shin","doi":"10.1016/j.jnt.2025.09.002","DOIUrl":"10.1016/j.jnt.2025.09.002","url":null,"abstract":"<div><div>For a positive integer <em>N</em>, we define an extended modular function of level <em>N</em> motivated by physics and investigate its fundamental properties. Let <em>K</em> be an imaginary quadratic field, and let <span><math><mi>O</mi></math></span> be an order in <em>K</em> of discriminant <em>D</em>. Let <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>O</mi><mo>,</mo><mspace></mspace><mi>N</mi></mrow></msub></math></span> denote the ray class field of <span><math><mi>O</mi></math></span> modulo <span><math><mi>N</mi><mi>O</mi></math></span>. For <span><math><mi>N</mi><mo>≥</mo><mn>3</mn></math></span>, we provide an explicit description of the Galois group <span><math><mrow><mi>Gal</mi></mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>O</mi><mo>,</mo><mspace></mspace><mi>N</mi></mrow></msub><mo>/</mo><mi>Q</mi><mo>)</mo></math></span> using special values of extended modular functions of level <em>N</em> and the definite form class group of discriminant <em>D</em> and level <em>N</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 808-824"},"PeriodicalIF":0.7,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267052","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":"Exceptional zero formulas for anticyclotomic p-adic L-functions","authors":"Víctor Hernández Barrios ,&nbsp;Santiago Molina Blanco","doi":"10.1016/j.jnt.2025.08.015","DOIUrl":"10.1016/j.jnt.2025.08.015","url":null,"abstract":"<div><div>In this note we define anticyclotomic <em>p</em>-adic measures attached to a modular elliptic curve <em>E</em> over a general number field <em>F</em>, a quadratic extension <span><math><mi>K</mi><mo>/</mo><mi>F</mi></math></span>, and a set of places <em>S</em> of <em>F</em> above <em>p</em>. We study the exceptional zero phenomenon that arises when <em>E</em> has multiplicative reduction at some place in <em>S</em>. In this direction, we obtain <em>p</em>-adic Gross-Zagier formulas relating derivatives of the corresponding <em>p</em>-adic L-functions to the extended Mordell-Weil group of <em>E</em>. Our main result uses the recent construction of plectic points on elliptic curves due to Fornea and Gehrmann and generalizes their main result in <span><span>[9]</span></span>. We obtain a formula that computes the <em>r</em>-th derivative of the <em>p</em>-adic L-function, where <em>r</em> is the number of places in <em>S</em> where <em>E</em> has multiplicative reduction, in terms of plectic points and Tate periods of <em>E</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 583-633"},"PeriodicalIF":0.7,"publicationDate":"2025-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267059","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 certain correlations into the divisor problem","authors":"Alexandre Dieguez","doi":"10.1016/j.jnt.2025.08.021","DOIUrl":"10.1016/j.jnt.2025.08.021","url":null,"abstract":"<div><div>For a fixed irrational <span><math><mi>θ</mi><mo>&gt;</mo><mn>0</mn></math></span> with a prescribed irrationality measure function, we study the correlation <span><math><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>1</mn></mrow><mrow><mi>X</mi></mrow></msubsup><mi>Δ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>Δ</mi><mo>(</mo><mi>θ</mi><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi></math></span>, where Δ is the Dirichlet error term in the divisor problem. When <em>θ</em> has a finite irrationality measure, it is known that decorrelation occurs at a rate expressible in terms of this measure. Strong decorrelation occurs for all positive irrationals, except possibly Liouville numbers. We show that for irrationals with a prescribed irrationality measure function <em>ψ</em>, decorrelation can be quantified in terms of <span><math><msup><mrow><mi>ψ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 519-536"},"PeriodicalIF":0.7,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220654","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":"The square-root law does not hold in the presence of zero divisors","authors":"Nathaniel Kingsbury-Neuschotz","doi":"10.1016/j.jnt.2025.08.020","DOIUrl":"10.1016/j.jnt.2025.08.020","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;em&gt;R&lt;/em&gt; be a finite ring (with identity, not necessarily commutative) and define the paraboloid &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Suppose that for a sequence of finite rings of size tending to infinity, the Fourier transform of &lt;em&gt;P&lt;/em&gt; satisfies a square-root law of the form &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ψ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; for all nontrivial additive characters &lt;em&gt;ψ&lt;/em&gt;, with &lt;em&gt;C&lt;/em&gt; some fixed constant (for instance, if &lt;em&gt;R&lt;/em&gt; is a finite field, this bound will be satisfied with &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;). Then all but finitely many of the rings are fields.&lt;/div&gt;&lt;div&gt;Most of our argument works in greater generality: let &lt;em&gt;f&lt;/em&gt; be a polynomial with integer coefficients in &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; variables, with a fixed order of variable multiplications (so that it defines a function &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&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;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; even when &lt;em&gt;R&lt;/em&gt; is noncommutative), and set &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. If (for a sequence of finite rings of size tending to infinity) we have a square root law for the Fourier transform of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, then all but finitely many of the rings are fields or matrix rings of small dimension. We also describe how our techniques can establish that certain varieties do not satisfy a square root law ","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 481-505"},"PeriodicalIF":0.7,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220605","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":"Spherical varieties and non-ordinary families of cohomology classes","authors":"Rob Rockwood","doi":"10.1016/j.jnt.2025.08.012","DOIUrl":"10.1016/j.jnt.2025.08.012","url":null,"abstract":"<div><div>We show that <em>p</em>-adic families of cohomology classes associated to symmetric spaces vary <em>p</em>-adically over small discs in weight space, without any ordinarity assumption. This generalises previous work of Loeffler, Zerbes and the author. Furthermore, we show that these families exhibit full variation in the cyclotomic direction, generalising previous constructions of Euler systems and <em>p</em>-adic <em>L</em>-functions. As an application we show that the Lemma–Flach Euler system of Loeffler–Skinner–Zerbes interpolates in Coleman families.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 390-454"},"PeriodicalIF":0.7,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220703","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":"Unconditional lower bounds for the sixth and eighth moments of the Riemann zeta function","authors":"Timothy Page","doi":"10.1016/j.jnt.2025.08.017","DOIUrl":"10.1016/j.jnt.2025.08.017","url":null,"abstract":"<div><div>Unconditional bounds on the sixth and eighth moments of the Riemann zeta function are improved by bounding twisted second and fourth moments that arise upon application of the Cauchy-Schwarz inequality and Hölder's inequality. An unconditional bound on the sixth moment of the derivative of the Riemann zeta function is also deduced.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 318-369"},"PeriodicalIF":0.7,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220651","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":"Higher moments for non-normal fields with Galois group Ad and Sd","authors":"Jiong Yang","doi":"10.1016/j.jnt.2025.08.018","DOIUrl":"10.1016/j.jnt.2025.08.018","url":null,"abstract":"<div><div>Let <em>K</em> be a non-normal number field of degree <em>d</em> with Galois group <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span>. Let <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of integral ideals of norm <em>n</em> in <em>K</em>. We obtain an asymptotic formula for the summation <span><math><munder><mo>∑</mo><mrow><mi>n</mi><mo>≤</mo><mi>x</mi></mrow></munder><msubsup><mrow><mi>a</mi></mrow><mrow><mi>K</mi></mrow><mrow><mi>l</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for any <span><math><mi>l</mi><mo>≥</mo><mn>2</mn></math></span>. As a consequence, we obtain such an asymptotic formula for any number field <em>K</em> of degree less or equal to 8 unconditionally.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 370-389"},"PeriodicalIF":0.7,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220652","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":"Explicit formulas for Grassmannian polylogarithms in weights 4 and 5","authors":"Steven Charlton ,&nbsp;Herbert Gangl ,&nbsp;Danylo Radchenko","doi":"10.1016/j.jnt.2025.08.011","DOIUrl":"10.1016/j.jnt.2025.08.011","url":null,"abstract":"<div><div>We explicitly reduce the Grassmannian polylogarithm in weight 4 and in weight 5 each to depth 2 iterated integrals. Furthermore, using this reduction in weight 4 we obtain an explicit, albeit complicated, form of the so-called 4-ratio, which gives an expression for the Borel class in continuous cohomology of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> in terms of <span><math><msub><mrow><mi>Li</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 537-582"},"PeriodicalIF":0.7,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220653","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 hook length biases of the 2- and 3-regular partitions","authors":"Wenxia Qu ,&nbsp;Wenston J.T. Zang","doi":"10.1016/j.jnt.2025.08.016","DOIUrl":"10.1016/j.jnt.2025.08.016","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the total number of <em>i</em> hooks in the <em>t</em>-regular partitions of <em>n</em>. Singh and Barman (2024) <span><span>[14]</span></span> raised two conjectures on <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. The first conjecture is on the positivity of <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>28</mn></math></span>. The second conjecture states that when <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for all <em>n</em> except for <span><math><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. In this paper, we confirm the first conjecture. Moreover, we show that for any odd <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, the second conjecture fails for infinitely many <em>n</em>. Furthermore, we verify that the second conjecture holds for <span><math><mi>k</mi><mo>=</mo><mn>4</mn></math></span> and 6. We also propose a conjecture on the even case <em>k</em>, which is a modification of Singh and Barman's second conjecture.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 455-480"},"PeriodicalIF":0.7,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220655","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":"Resolution of Erdős' problems about unimodularity","authors":"Stijn Cambie","doi":"10.1016/j.jnt.2025.08.014","DOIUrl":"10.1016/j.jnt.2025.08.014","url":null,"abstract":"<div><div>Letting <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> be the density of the set of integers with exactly one divisor in <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span>, Erdős wondered if <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> is unimodular for fixed <em>n</em>. We prove this is false in general, as the sequence <span><math><mo>(</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>)</mo></math></span> has superpolynomially many local extrema. However, we confirm unimodality in the single case for which it occurs; <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span>. We also solve the question on unimodality of the density of integers whose <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>t</mi><mi>h</mi></mrow></msup></math></span> prime is <em>p</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 271-277"},"PeriodicalIF":0.7,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145158355","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}