We prove quantitative estimates for averages of the von Mangoldt and Möbius functions along polynomial progressions for a large class of polynomials . The error terms obtained save an arbitrary power of logarithm, matching the classical Siegel–Walfisz error term. These results give the first quantitative bounds for the Tao–Ziegler polynomial patterns in the primes result. The proofs are based on a quantitative generalised von Neumann theorem of Peluse, a recent result of Leng on strong bounds for the Gowers uniformity of the primes, and analysis of a ‘Siegel model’ for the von Mangoldt function along polynomial progressions.
{"title":"Quantitative asymptotics for polynomial patterns in the primes","authors":"Lilian Matthiesen, Joni Teräväinen, Mengdi Wang","doi":"10.1112/mtk.70103","DOIUrl":"https://doi.org/10.1112/mtk.70103","url":null,"abstract":"<p>We prove quantitative estimates for averages of the von Mangoldt and Möbius functions along polynomial progressions <span></span><math></math> for a large class of polynomials <span></span><math></math>. The error terms obtained save an arbitrary power of logarithm, matching the classical Siegel–Walfisz error term. These results give the first quantitative bounds for the Tao–Ziegler polynomial patterns in the primes result. The proofs are based on a quantitative generalised von Neumann theorem of Peluse, a recent result of Leng on strong bounds for the Gowers uniformity of the primes, and analysis of a ‘Siegel model’ for the von Mangoldt function along polynomial progressions.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70103","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148048156","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}
We consider equations of the form where the variables are all taken to be primes. We define an analogue of the Hasse principle for solubility in the primes (which we call the prime Hasse principle), and prove that, whenever , , this holds for almost all such equations. This is based on work of Brüdern and Dietmann on the Hasse principle. We then prove some further results about prime solubility and the prime Hasse principle, including a partial converse, and some counterexamples. Of particular interest are counterexamples of degree 2, which show that the analogue of the Hasse–Minkowski theorem fails for prime solubility.
{"title":"Random Diophantine equations in the primes","authors":"Philippa Holdridge","doi":"10.1112/mtk.70102","DOIUrl":"https://doi.org/10.1112/mtk.70102","url":null,"abstract":"<p>We consider equations of the form <span></span><math></math> where the variables <span></span><math></math> are all taken to be primes. We define an analogue of the Hasse principle for solubility in the primes (which we call the prime Hasse principle), and prove that, whenever <span></span><math></math>, <span></span><math></math>, this holds for almost all such equations. This is based on work of Brüdern and Dietmann on the Hasse principle. We then prove some further results about prime solubility and the prime Hasse principle, including a partial converse, and some counterexamples. Of particular interest are counterexamples of degree 2, which show that the analogue of the Hasse–Minkowski theorem fails for prime solubility.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70102","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148047448","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}
Let be a real quadratic number field, and let denote its cyclotomic -extension. For each integer , let be the unique intermediate field in such that . By studying the 2-adic divisibility of Dirichlet -series at negative integers, we derive an asymptotic formula that determines the order of the 2-primary part of even -groups of rings of integers of for sufficiently large . As a corollary, we determine their and invariants. We also establish a lower bound for beyond which this asymptotic formula holds. Our results have two main applications: (1) For , or with , we determine the structure of the 2-primary tame kernels . (2) We explicitly determine the three Iwasawa invariants for a family of real quadratic number fields, whose discriminants have arbitrarily many prime divisors.
{"title":"Iwasawa invariants of even -groups of rings of integers in the -extension over real quadratic number fields","authors":"Li-Tong Deng, Yong-Xiong Li","doi":"10.1112/mtk.70100","DOIUrl":"https://doi.org/10.1112/mtk.70100","url":null,"abstract":"<p>Let <span></span><math></math> be a real quadratic number field, and let <span></span><math></math> denote its cyclotomic <span></span><math></math>-extension. For each integer <span></span><math></math>, let <span></span><math></math> be the unique intermediate field in <span></span><math></math> such that <span></span><math></math>. By studying the 2-adic divisibility of Dirichlet <span></span><math></math>-series at negative integers, we derive an asymptotic formula that determines the order of the 2-primary part of even <span></span><math></math>-groups of rings of integers of <span></span><math></math> for sufficiently large <span></span><math></math>. As a corollary, we determine their <span></span><math></math> and <span></span><math></math> invariants. We also establish a lower bound for <span></span><math></math> beyond which this asymptotic formula holds. Our results have two main applications: (1) For <span></span><math></math>, <span></span><math></math> or <span></span><math></math> with <span></span><math></math>, we determine the structure of the 2-primary tame kernels <span></span><math></math>. (2) We explicitly determine the three Iwasawa invariants <span></span><math></math> for a family of real quadratic number fields, whose discriminants have arbitrarily many prime divisors.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148050740","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}
We establish an asymptotic formula for counting integer solutions with smooth weights to an equation of the form , where is a non-zero integer, with an explicit main term and a strong bound on the error term in terms of the size of the variables as well as of . We also establish an asymptotic formula for counting integer solutions with smooth weights to the congruence , where is a large prime, with a strong bound on the error term.
{"title":"Distribution of integer points on determinant surfaces and a mod-p analogue","authors":"Satadal Ganguly, Rachita Guria","doi":"10.1112/mtk.70093","DOIUrl":"10.1112/mtk.70093","url":null,"abstract":"<p>We establish an asymptotic formula for counting integer solutions with smooth weights to an equation of the form <span></span><math></math>, where <span></span><math></math> is a non-zero integer, with an explicit main term and a strong bound on the error term in terms of the size of the variables <span></span><math></math> as well as of <span></span><math></math>. We also establish an asymptotic formula for counting integer solutions with smooth weights to the congruence <span></span><math></math>, where <span></span><math></math> is a large prime, with a strong bound on the error term.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70093","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147683708","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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