Journal of Number Theory最新文献

{"title":"Erdős inequality for primitive sets","authors":"Petr Kucheriaviy","doi":"10.1016/j.jnt.2025.08.004","DOIUrl":"10.1016/j.jnt.2025.08.004","url":null,"abstract":"<div><div>A set of natural numbers <em>A</em> is called primitive if no element of <em>A</em> divides any other. Let <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of prime divisors of <em>n</em> counted with multiplicity. Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup></mrow><mrow><mi>a</mi><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>a</mi><mo>)</mo></mrow><mrow><mi>z</mi></mrow></msup></mrow></mfrac></math></span>, where <span><math><mi>z</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></math></span>. Erdős proved in 1935 that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi><mi>log</mi><mo>⁡</mo><mi>a</mi></mrow></mfrac></math></span> is uniformly bounded over all primitive sets <em>A</em>. We prove a generalization of Erdős inequality which provides an analogous result for <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, when <span><math><mi>z</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. Furthermore, we study the supremum of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> over all primitive sets. We also discuss <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>z</mi><mo>→</mo><mn>0</mn></mrow></msub><mo>⁡</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, which is a generalization of Dirichlet density. We study the asymptotics of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>n</mi><mo>:</mo><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>k</mi><mo>}</mo></math></span>. For <span><math><mi>z</mi><mo>=</mo><mn>1</mn></math></span> we find the next term in asymptotic expansion of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> refining the result of Gorodetsky, Lichtman, and Wong. We also study the supremum of <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup><mo>/</mo><mi>a</mi></math></span> over all primitive subsets of <span><math><mo>[</mo><mn>1","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 113-152"},"PeriodicalIF":0.7,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096121","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 minimal denominator problem in function fields","authors":"Noy Soffer Aranov","doi":"10.1016/j.jnt.2025.08.005","DOIUrl":"10.1016/j.jnt.2025.08.005","url":null,"abstract":"<div><div>We study the minimal denominator problem in function fields. In particular, we compute the probability distribution function of the random variable which returns the degree of the smallest denominator <em>Q</em>, for which the ball of a fixed radius around a point contains a rational function of the form <span><math><mfrac><mrow><mi>P</mi></mrow><mrow><mi>Q</mi></mrow></mfrac></math></span>. Moreover, we discuss the distribution of the random variable which returns the denominator of minimal degree, as well as higher dimensional and <em>P</em>-adic generalizations. This can be viewed as a function field generalization of a paper by Chen and Haynes.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 35-48"},"PeriodicalIF":0.7,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049555","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":"Murmurations and Sato–Tate conjectures for high rank zetas of elliptic curves","authors":"Zhan Shi ,&nbsp;Lin Weng","doi":"10.1016/j.jnt.2025.07.006","DOIUrl":"10.1016/j.jnt.2025.07.006","url":null,"abstract":"<div><div>For elliptic curves over rationals, there are a well-known conjecture of Sato–Tate and a new computational guided murmuration phenomenon, for which the abelian Artin zeta functions are used. In this paper, we show that both the murmurations and the Sato–Tate conjecture stand equally well for non-abelian high rank zeta functions of the <em>p</em>-reductions of elliptic curves over rationals. We establish our results by carefully examining asymptotic behaviors of the <em>p</em>-reduction invariants <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><mi>n</mi></mrow></msub><mspace></mspace><mo>(</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>)</mo></math></span>, the rank <em>n</em> analogous of the rank one <em>a</em>-invariant <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msub><mo>=</mo><mn>1</mn><mo>+</mo><mi>p</mi><mo>−</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msub></math></span> of elliptic curve <span><math><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>. Such asymptotic results are based on a ‘counting miracle’ of the so-called <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mi>n</mi></mrow></msub></math></span>- and <span><math><msub><mrow><mi>β</mi></mrow><mrow><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mi>n</mi></mrow></msub></math></span>-invariants of <span><math><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> in rank <em>n</em>, and a remarkable recursive relation on the <span><math><msub><mrow><mi>β</mi></mrow><mrow><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mi>n</mi></mrow></msub></math></span>-invariants, both established by Weng–Zagier in <span><span>[22]</span></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 948-968"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144922301","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":"Invariant part of class groups and distribution of relative class group","authors":"Weitong Wang","doi":"10.1016/j.jnt.2025.07.012","DOIUrl":"10.1016/j.jnt.2025.07.012","url":null,"abstract":"<div><div>We generalize the work of Roquette and Zassenhaus on the invariant part of the class groups to the relative class groups. Thus, we can show some statistical results as follows. For abelian extensions over a fixed number field <em>K</em>, we show infinite <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-moments for the Sylow <em>p</em>-subgroup of the relative class group when <em>p</em> divides the degree of the extension. For sextic number fields with <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-closure, we can show infinite <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-moments for the Sylow 2-subgroup of the relative class group when the extensions run over a fixed Galois cubic field.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 691-748"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904483","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":"Goldbach representations with several primes","authors":"Thi Thu Nguyen","doi":"10.1016/j.jnt.2025.07.008","DOIUrl":"10.1016/j.jnt.2025.07.008","url":null,"abstract":"<div><div>We study an asymptotic formula for average orders of Goldbach representations of an integer as the sum of <em>k</em> primes. We extend the existing result for <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span> to a general <em>k</em> and obtain a better error term for all <em>k</em> larger than 3. Moreover, we prove an equivalence between the Riemann Hypothesis and a good average order in this case.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 858-877"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144908191","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":"Numerical investigation of lower order biases in moment expansions of one parameter families of elliptic curves","authors":"Timothy Cheek ,&nbsp;Pico Gilman ,&nbsp;Kareem Jaber ,&nbsp;Steven J. Miller ,&nbsp;Vismay Sharan ,&nbsp;Marie-Hélène Tomé","doi":"10.1016/j.jnt.2025.07.003","DOIUrl":"10.1016/j.jnt.2025.07.003","url":null,"abstract":"<div><div>For a fixed elliptic curve <em>E</em> without complex multiplication, <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>≔</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>#</mi><mi>E</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> is <span><math><mi>O</mi><mo>(</mo><msqrt><mrow><mi>p</mi></mrow></msqrt><mo>)</mo></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>/</mo><mn>2</mn><msqrt><mrow><mi>p</mi></mrow></msqrt></math></span> converges to a semicircular distribution. Michel proved that for a one-parameter family of elliptic curves <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><mo>(</mo><mi>T</mi><mo>)</mo><mi>x</mi><mo>+</mo><mi>B</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> with <span><math><mi>A</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>,</mo><mi>B</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>T</mi><mo>]</mo></math></span> and non-constant <em>j</em>-invariant, the second moment of <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span>. The size and sign of the lower order terms has applications to the distribution of zeros near the central point of Hasse-Weil <em>L</em>-functions and the Birch and Swinnerton-Dyer conjecture. S. J. Miller conjectured that the highest order term of the lower order terms of the second moment that does not average to zero is on average negative. Previous work on the conjecture has been restricted to a small set of highly nongeneric families. We create a database and a framework to quickly and systematically investigate biases in the second moment of any one-parameter family. When looking at families which have so far been beyond current theory, we find several potential violations of the conjecture for <span><math><mi>p</mi><mo>≤</mo><mn>250</mn><mo>,</mo><mn>000</mn></math></span> and discuss new conjectures motivated by the data.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 929-947"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144912893","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":"Elasticity of orders with prime conductor","authors":"Jared Kettinger ,&nbsp;Grant Moles","doi":"10.1016/j.jnt.2025.07.013","DOIUrl":"10.1016/j.jnt.2025.07.013","url":null,"abstract":"<div><div>Let <em>R</em> be an order in a number field whose conductor ideal <span><math><mi>P</mi><mo>:</mo><mo>=</mo><mo>(</mo><mi>R</mi><mo>:</mo><mover><mrow><mi>R</mi></mrow><mo>‾</mo></mover><mo>)</mo></math></span> is prime in the ring of integers <span><math><mover><mrow><mi>R</mi></mrow><mo>‾</mo></mover></math></span>. In this paper, we explore the factorization properties of such orders. Most notably, we give a complete characterization of the elasticity of <em>R</em> in terms of its class group. We conclude with an application to the computation of class groups of certain orders.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 579-593"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144887133","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":"New bounds in R.S. Lehman's estimates for the difference π(x)−li(x)","authors":"Michael Revers","doi":"10.1016/j.jnt.2025.07.002","DOIUrl":"10.1016/j.jnt.2025.07.002","url":null,"abstract":"<div><div>We denote by <span><math><mi>π</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math></span> the usual prime counting function and let <span><math><mi>l</mi><mi>i</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math></span> the logarithmic integral of <em>x</em>. In 1966, R.S. Lehman came up with a new approach and an effective method for finding an upper bound where it is assured that a sign change occurs for <span><math><mi>π</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mi>l</mi><mi>i</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math></span> for some value <em>x</em> not higher than this given bound. In this paper we provide further improvements on the error terms including an improvement upon Lehman's famous error term <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> in his original paper. We are now able to completely eliminate the lower condition for the size-length <em>η</em>. For further numerical computations this enables us to establish sharper results on the positions for the sign changes. We illustrate with some numerical computations on the lowest known crossover regions near 10³¹⁶ and we discuss numerically on potential crossover regions below this value.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 878-909"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144908203","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":"Note on a theorem of Birch–Erdős and m-ary partitions","authors":"Yuchen Ding ,&nbsp;Honghu Liu ,&nbsp;Zi Wang","doi":"10.1016/j.jnt.2025.07.009","DOIUrl":"10.1016/j.jnt.2025.07.009","url":null,"abstract":"<div><div>Let <span><math><mi>p</mi><mo>,</mo><mi>q</mi><mo>&gt;</mo><mn>1</mn></math></span> be two relatively prime integers and <span><math><mi>N</mi></math></span> the set of nonnegative integers. Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of different expressions of <em>n</em> written as a sum of distinct terms taken from <span><math><mo>{</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mi>q</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>:</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math></span>. Erdős conjectured and then Birch proved that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><mn>1</mn></math></span> provided that <em>n</em> is sufficiently large. In this note, for all sufficiently large number <em>n</em> we prove<span><span><span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mfrac><mrow><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><mi>log</mi><mo>⁡</mo><mi>p</mi><mi>log</mi><mo>⁡</mo><mi>q</mi></mrow></mfrac><mo>(</mo><mn>1</mn><mo>+</mo><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>/</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow></msup><mo>.</mo></math></span></span></span> We also show that <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mo>⁡</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Additionally, we will point out the relations between <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and <em>m</em>-ary partitions.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 910-928"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144908202","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":"Variants of Kohnen's conjecture for Hermitian modular forms","authors":"Biplab Paul ,&nbsp;Sujeet Kumar Singh","doi":"10.1016/j.jnt.2025.07.001","DOIUrl":"10.1016/j.jnt.2025.07.001","url":null,"abstract":"<div><div>Let <em>F</em> be a Hermitian cusp form of weight <em>k</em> and of degree 2 over <span><math><mi>Q</mi><mo>(</mo><mi>i</mi><mo>)</mo></math></span> with Fourier-Jacobi coefficients <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>, <span><math><mi>m</mi><mo>∈</mo><mi>N</mi></math></span>. Motivated by a conjecture of W. Kohnen on the growth of Petersson norm of <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> in the set-up of Siegel modular forms, we study analogous questions in the set-up of Hermitian modular forms. We first propose a conjecture in this set-up which is analogous to that of Kohnen. We then provide some evidence by proving the conjecture for cusp forms lying in the Hermitian-Maass subspace. We also study certain other related problems.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 626-650"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904481","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}