{"title":"On prime numbers and quadratic forms represented by positive-definite, primitive quadratic forms","authors":"Yves Martin","doi":"10.1016/j.jnt.2024.12.014","DOIUrl":"10.1016/j.jnt.2024.12.014","url":null,"abstract":"<div><div>In this note we show that every positive-definite, integral, primitive, <em>n</em>-ary quadratic form with <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> represents infinitely many prime numbers and infinitely many primitive, non-equivalent, <em>m</em>-ary quadratic forms for each <span><math><mn>2</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>. We do so via an inductive argument which only requires to know the statement for <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> (proved by H. Weber in 1882), and elementary linear algebra. The result on the representation of prime numbers by <em>n</em>-ary quadratic forms for arbitrary <span><math><mi>n</mi><mo>></mo><mn>2</mn></math></span> can be deduced from theorems already known, but the proof below is more direct and seems to be new in the literature. As an application we establish a non-vanishing result for Fourier-Jacobi coefficients of Siegel modular forms of any degree, level and Dirichlet character, subject to a condition on the conductor of the character.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 26-36"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143520773","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":"Uniform bounds for Kloosterman sums of half-integral weight, same-sign case","authors":"Qihang Sun","doi":"10.1016/j.jnt.2024.11.012","DOIUrl":"10.1016/j.jnt.2024.11.012","url":null,"abstract":"<div><div>In the previous paper <span><span>[Sun24]</span></span>, the author proved a uniform bound for sums of half-integral weight Kloosterman sums. This bound was applied to prove an exact formula for partitions of rank modulo 3. That uniform estimate provides a more precise bound for a certain class of multipliers compared to the 1983 result by Goldfeld and Sarnak and generalizes the 2009 result from Sarnak and Tsimerman to the half-integral weight case. However, the author only considered the case when the parameters satisfied <span><math><mover><mrow><mi>m</mi></mrow><mrow><mo>˜</mo></mrow></mover><mover><mrow><mi>n</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo><</mo><mn>0</mn></math></span>. In this paper, we prove the same uniform bound when <span><math><mover><mrow><mi>m</mi></mrow><mrow><mo>˜</mo></mrow></mover><mover><mrow><mi>n</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>></mo><mn>0</mn></math></span> for further applications.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 104-139"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143548347","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 Mahler measure of a family of polynomials with arbitrarily many variables","authors":"Siva Sankar Nair","doi":"10.1016/j.jnt.2024.11.011","DOIUrl":"10.1016/j.jnt.2024.11.011","url":null,"abstract":"<div><div>We present an exact formula for the Mahler measure of an infinite family of polynomials with arbitrarily many variables. The formula is obtained by manipulating the integral defining the Mahler measure using certain transformations, followed by an iterative process that reduces this computation to the evaluation of certain polylogarithm functions at sixth roots of unity. This yields values of the Riemann zeta function and the Dirichlet <em>L</em>-function associated to the character of conductor 3.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 214-272"},"PeriodicalIF":0.6,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143637189","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":"Relative sizes of iterated sumsets","authors":"Noah Kravitz","doi":"10.1016/j.jnt.2025.01.007","DOIUrl":"10.1016/j.jnt.2025.01.007","url":null,"abstract":"<div><div>Let <em>hA</em> denote the <em>h</em>-fold sumset of a subset <em>A</em> of an abelian group. Resolving a problem of Nathanson, we show that for any prescribed permutations <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, there exist finite subsets <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊆</mo><mi>Z</mi></math></span> such that for each <span><math><mn>1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><mi>H</mi></math></span>, the relative order of the quantities <span><math><mo>|</mo><mi>h</mi><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>,</mo><mo>…</mo><mo>,</mo><mo>|</mo><mi>h</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>|</mo></math></span> is given by <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span>. We also establish extensions where <span><math><mi>Z</mi></math></span> is replaced by any other infinite abelian group or where one prescribes some equalities (not only inequalities) among the sumset sizes.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"272 ","pages":"Pages 113-128"},"PeriodicalIF":0.6,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510120","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":"Upper bounds on large deviations of Dirichlet L-functions in the q-aspect","authors":"Louis-Pierre Arguin, Nathan Creighton","doi":"10.1016/j.jnt.2025.01.009","DOIUrl":"10.1016/j.jnt.2025.01.009","url":null,"abstract":"<div><div>We prove a result on the large deviations of the central values of even primitive Dirichlet <em>L</em>-functions with a given modulus. For <span><math><mi>V</mi><mo>∼</mo><mi>α</mi><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi></math></span> with <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></math></span>, we show that<span><span><span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>φ</mi><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mfrac><mi>#</mi><mrow><mo>{</mo><mi>χ</mi><mtext> even, primitive mod </mtext><mi>q</mi><mo>:</mo><mi>log</mi><mo></mo><mrow><mo>|</mo><mi>L</mi><mrow><mo>(</mo><mi>χ</mi><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>|</mo></mrow><mo>></mo><mi>V</mi><mo>}</mo></mrow><mspace></mspace><mo>≪</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mfrac><mrow><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi></mrow></mfrac></mrow></msup></mrow><mrow><msqrt><mrow><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi></mrow></msqrt></mrow></mfrac><mo>.</mo></math></span></span></span> This yields the sharp upper bound for the fractional moments of central values of Dirichlet <em>L</em>-functions proved by Gao, upon noting that the number of even, primitive characters with modulus <em>q</em> is <span><math><mfrac><mrow><mi>φ</mi><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. The proof is an adaptation to the <em>q</em>-aspect of the recursive scheme developed by Arguin, Bourgade and Radziwiłł for the local maxima of the Riemann zeta function, and applied by Arguin and Bailey to the large deviations in the <em>t</em>-aspect. We go further and get bounds on the case where <span><math><mi>V</mi><mo>=</mo><mi>o</mi><mo>(</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi><mo>)</mo></math></span>. These bounds are not expected to be sharp, but the discrepancy from the Central Limit Theorem estimate grows very slowly with <em>q</em>. The method involves a formula for the twisted mollified second moment of central values of Dirichlet <em>L</em>-functions, building on the work of Iwaniec and Sarnak.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"273 ","pages":"Pages 96-158"},"PeriodicalIF":0.6,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511909","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":"Square patterns in dynamical orbits","authors":"Vefa Goksel , Giacomo Micheli","doi":"10.1016/j.jnt.2024.12.004","DOIUrl":"10.1016/j.jnt.2024.12.004","url":null,"abstract":"<div><div>Let <em>q</em> be an odd prime power. Let <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> be a polynomial having degree at least 2, <span><math><mi>a</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, and denote by <span><math><msup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> the <em>n</em>-th iteration of <em>f</em>. Let <em>χ</em> be the quadratic character of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, and <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>)</mo></math></span> the forward orbit of <em>a</em> under iteration by <em>f</em>. Suppose that the sequence <span><math><msub><mrow><mo>(</mo><mi>χ</mi><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>a</mi><mo>)</mo><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></math></span> is periodic, and <em>m</em> is its period. Assuming a mild and generic condition on <em>f</em>, we show that, up to a constant depending on <em>d</em>, <em>m</em> can be bounded from below by <span><math><mo>|</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>)</mo><mo>|</mo><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mfrac><mrow><mn>2</mn><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mo>(</mo><mi>d</mi><mo>)</mo><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mo>(</mo><mi>d</mi><mo>)</mo><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></msup></math></span> as <em>q</em> grows. More informally, we prove that the period of the appearance of squares in an orbit of an element provides an upper bound for the size of the orbit itself. Using a similar method, we can also prove that, up to a constant depending on <em>d</em>, we cannot have more than <span><math><msup><mrow><mi>q</mi></mrow><mrow><mfrac><mrow><mn>2</mn><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mo>(</mo><mi>d</mi><mo>)</mo><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mo>(</mo><mi>d</mi><mo>)</mo><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></msup></math></span> consecutive squares or non-squares in the forward orbit of <em>a</em>. In addition, using geometric tools from global function field theory such as abc theorem, we provide a classification of all polynomials for which our generic condition does not hold, making the results effective. Interestingly enough, our condition is purely geometrical, while our final results are completely arithmetical. As a corollary, this paper removes most of the hypothesis of (Ostafe, Shparlinski. Proceedings of the American Mathematical Society 138.8 (2010)), most notably extending the results to an","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"272 ","pages":"Pages 129-146"},"PeriodicalIF":0.6,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143512223","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":"An asymptotic formula involving the triple divisor function","authors":"Guangwei Hu , Chenran Xu","doi":"10.1016/j.jnt.2025.01.011","DOIUrl":"10.1016/j.jnt.2025.01.011","url":null,"abstract":"<div><div>Suppose <span><math><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denotes the classical triple divisor function. Let <span><math><mi>Q</mi><mo>(</mo><mrow><mi>x</mi><mo>)</mo></mrow></math></span> be a positive definite integral quadratic form, and <span><math><mi>r</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> denote the number of representations of <em>n</em> by the quadratic form <em>Q</em>. In this paper, we will establish an asymptotic formula of the summation<span><span><span><math><munder><mo>∑</mo><mrow><mi>n</mi><mo>≤</mo><mi>X</mi></mrow></munder><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>+</mo><mi>h</mi><mo>)</mo><mi>r</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>Q</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <em>h</em> is a positive integer satisfying <span><math><mi>h</mi><mo>≤</mo><mi>H</mi><mo>≪</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>ε</mi></mrow></msup></math></span>. Our result breaks through the trivial bound of the above summation and obtains the power saving in <em>O</em>-term.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 328-347"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445484","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":"Some improvements on the Davenport-Heilbronn method","authors":"Konstantinos Kydoniatis","doi":"10.1016/j.jnt.2025.01.013","DOIUrl":"10.1016/j.jnt.2025.01.013","url":null,"abstract":"<div><div>Let <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, <span><math><mi>s</mi><mo>≥</mo><mo>⌈</mo><mi>k</mi><mo>(</mo><mi>log</mi><mo></mo><mi>k</mi><mo>+</mo><mn>4.20032</mn><mo>)</mo><mo>⌉</mo></math></span>, and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mi>ω</mi><mo>∈</mo><mi>R</mi></math></span>. Assume that the <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> are non-zero, not all in rational ratio, and not all of the same sign in the case that <em>k</em> is even. Then, for any <span><math><mi>ϵ</mi><mo>></mo><mn>0</mn></math></span>, the inequality<span><span><span><math><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>s</mi></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>+</mo><mi>ω</mi><mo>|</mo><mo><</mo><mi>ϵ</mi></math></span></span></span> has <span><math><mo>≫</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>s</mi><mo>−</mo><mi>k</mi></mrow></msup></math></span> integer solutions with <span><math><mo>|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>≤</mo><mi>P</mi></math></span>. Moreover the asymptotic formula for the number of smooth solutions is established assuming the same conditions hold.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"272 ","pages":"Pages 1-17"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474666","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":"Heights of rational points on Mordell curves","authors":"Alan Zhao","doi":"10.1016/j.jnt.2025.01.012","DOIUrl":"10.1016/j.jnt.2025.01.012","url":null,"abstract":"<div><div>We conjecture a lower bound for the minimal canonical height of non-torsion rational points on a natural density 1 subset of the sextic twist family of Mordell curves. We then establish a lower bound that yields a partial result towards this conjecture.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"272 ","pages":"Pages 18-33"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474667","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":"Negative first moment of quadratic twists of L-functions","authors":"Peng Gao , Liangyi Zhao","doi":"10.1016/j.jnt.2025.01.003","DOIUrl":"10.1016/j.jnt.2025.01.003","url":null,"abstract":"<div><div>We evaluate asymptotically the negative first moment at points larger than 1/2 of the family of quadratic twists of automorphic <em>L</em>-functions using multiple Dirichlet series under the generalized Riemann hypothesis and the Ramanujan-Petersson conjecture.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 389-406"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445486","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}
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