Pub Date : 2026-07-01Epub Date: 2026-02-04DOI: 10.1016/j.jnt.2025.12.010
Changsong Shi, Liuquan Wang
Around 2016, Calinescu, Milas and Penn conjectured that the rank r Nahm sum associated with the tadpole Cartan matrix is modular, and they provided a proof for . The case was recently resolved by Milas and Wang. We prove this conjecture for the next cases . We also prove the modularity of some companion Nahm sums by establishing the corresponding Rogers–Ramanujan type identities. A key new ingredient in our proofs is some rank reduction formulas which allow us to decompose higher rank tadpole Nahm sums to mixed products of some lower rank Nahm-type sums and theta functions.
{"title":"Modularity of tadpole Nahm sums in ranks 4 and 5","authors":"Changsong Shi, Liuquan Wang","doi":"10.1016/j.jnt.2025.12.010","DOIUrl":"10.1016/j.jnt.2025.12.010","url":null,"abstract":"<div><div>Around 2016, Calinescu, Milas and Penn conjectured that the rank <em>r</em> Nahm sum associated with the <span><math><mi>r</mi><mo>×</mo><mi>r</mi></math></span> tadpole Cartan matrix is modular, and they provided a proof for <span><math><mi>r</mi><mo>=</mo><mn>2</mn></math></span>. The <span><math><mi>r</mi><mo>=</mo><mn>3</mn></math></span> case was recently resolved by Milas and Wang. We prove this conjecture for the next cases <span><math><mi>r</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn></math></span>. We also prove the modularity of some companion Nahm sums by establishing the corresponding Rogers–Ramanujan type identities. A key new ingredient in our proofs is some rank reduction formulas which allow us to decompose higher rank tadpole Nahm sums to mixed products of some lower rank Nahm-type sums and theta functions.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 214-245"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190614","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}
Pub Date : 2026-07-01Epub Date: 2026-02-03DOI: 10.1016/j.jnt.2026.01.001
Si Duc Quang
In this paper, we give a finiteness criterion for the solutions of the sequence of semi-q-decomposable form equations and inequalities, where the semi-q-decomposable form is factorized into a family of q nonconstant homogeneous polynomials with the distributive constant not exceeding a certain number.
{"title":"Finiteness criteria for the solutions of a sequence of decomposable form inequalities","authors":"Si Duc Quang","doi":"10.1016/j.jnt.2026.01.001","DOIUrl":"10.1016/j.jnt.2026.01.001","url":null,"abstract":"<div><div>In this paper, we give a finiteness criterion for the solutions of the sequence of semi-<em>q</em>-decomposable form equations and inequalities, where the semi-<em>q</em>-decomposable form is factorized into a family of <em>q</em> nonconstant homogeneous polynomials with the distributive constant not exceeding a certain number.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 131-148"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190617","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}
Pub Date : 2026-07-01Epub Date: 2026-01-20DOI: 10.1016/j.jnt.2025.12.004
Thomas H. Geisser
We give a version of the Artin-Tate formula for surfaces over a finite field which holds without assuming Tate's conjecture for divisors. The formula gives an equality between terms related to the Brauer group on the one hand, and terms related to the Néron-Severi group on the other hand. We give estimates on the terms appearing in the formula and use this to give sharp estimates on the size of the Brauer group of abelian surfaces depending on the p-rank.
{"title":"Brauer and Néron-Severi groups of surfaces over finite fields","authors":"Thomas H. Geisser","doi":"10.1016/j.jnt.2025.12.004","DOIUrl":"10.1016/j.jnt.2025.12.004","url":null,"abstract":"<div><div>We give a version of the Artin-Tate formula for surfaces over a finite field which holds without assuming Tate's conjecture for divisors. The formula gives an equality between terms related to the Brauer group on the one hand, and terms related to the Néron-Severi group on the other hand. We give estimates on the terms appearing in the formula and use this to give sharp estimates on the size of the Brauer group of abelian surfaces depending on the <em>p</em>-rank.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 38-61"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190681","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}
Pub Date : 2026-07-01Epub Date: 2026-01-21DOI: 10.1016/j.jnt.2025.12.005
Bruno Aguiló-Vidal , Luis Arenas-Carmona , Matías Saavedra-Lagos
We apply the theory of branches in Bruhat-Tits trees, developed in previous works by the second author and others, to the study of two dimensional representations of finite groups over the ring of integers of a number field. We provide a general strategy to perform these computations, and we give explicit formulas for some particular families.
{"title":"Two dimensional integral representations via branches of the Bruhat-Tits tree","authors":"Bruno Aguiló-Vidal , Luis Arenas-Carmona , Matías Saavedra-Lagos","doi":"10.1016/j.jnt.2025.12.005","DOIUrl":"10.1016/j.jnt.2025.12.005","url":null,"abstract":"<div><div>We apply the theory of branches in Bruhat-Tits trees, developed in previous works by the second author and others, to the study of two dimensional representations of finite groups over the ring of integers of a number field. We provide a general strategy to perform these computations, and we give explicit formulas for some particular families.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 62-103"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190618","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}
Pub Date : 2026-07-01Epub Date: 2026-02-03DOI: 10.1016/j.jnt.2025.12.012
Wu-Xia Ma , Yong-Gao Chen
Text
Let W be a non-empty subset of . A set is called an additive complement to W if . An additive complement C is said to be minimal if no proper subset of C is an additive complement to W. In 2019, Kiss, Sándor, and Yang proved that if m is a positive integer and , where X and Y are finite sets that satisfy certain conditions, then W has a minimal additive complement. They asked whether this is true if Y is an infinite set and does not contain an arithmetic progression of common difference m. In this paper, we present some results on this problem and pose three open problems for further research.
Video
For a video summary of this paper, please visit https://youtu.be/eSZCqS99au0.
设W为Z的一个非空子集。当集C+W=Z时，称集C≠Z为W的加性补。在2019年，Kiss, Sándor， and Yang证明了如果m是正整数且W=(mN+X)∪Y，其中X和Y是满足一定条件的有限集，则W具有最小的可加补。他们问，如果Y是一个无限集，并且不包含公差m的等差数列，这是否成立。在本文中，我们给出了这个问题的一些结果，并提出了三个开放的问题供进一步研究。观看本文的视频摘要，请访问https://youtu.be/eSZCqS99au0。
{"title":"On a problem of minimal additive complements of integers","authors":"Wu-Xia Ma , Yong-Gao Chen","doi":"10.1016/j.jnt.2025.12.012","DOIUrl":"10.1016/j.jnt.2025.12.012","url":null,"abstract":"<div><h3>Text</h3><div>Let <em>W</em> be a non-empty subset of <span><math><mi>Z</mi></math></span>. A set <span><math><mi>C</mi><mo>⊆</mo><mi>Z</mi></math></span> is called an additive complement to <em>W</em> if <span><math><mi>C</mi><mo>+</mo><mi>W</mi><mo>=</mo><mi>Z</mi></math></span>. An additive complement <em>C</em> is said to be minimal if no proper subset of <em>C</em> is an additive complement to <em>W</em>. In 2019, Kiss, Sándor, and Yang proved that if <em>m</em> is a positive integer and <span><math><mi>W</mi><mo>=</mo><mo>(</mo><mi>m</mi><mi>N</mi><mo>+</mo><mi>X</mi><mo>)</mo><mo>∪</mo><mi>Y</mi></math></span>, where <em>X</em> and <em>Y</em> are finite sets that satisfy certain conditions, then <em>W</em> has a minimal additive complement. They asked whether this is true if <em>Y</em> is an infinite set and does not contain an arithmetic progression of common difference <em>m</em>. In this paper, we present some results on this problem and pose three open problems for further research.</div></div><div><h3>Video</h3><div>For a video summary of this paper, please visit <span><span>https://youtu.be/eSZCqS99au0</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 178-187"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190616","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}
Pub Date : 2026-07-01Epub Date: 2026-01-21DOI: 10.1016/j.jnt.2025.12.008
Aidan Botkin , Madeline L. Dawsey , David J. Hemmer , Matthew R. Just , Robert Schneider
We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that naturally predicts the prime number theorem as well as the twin prime conjecture. The model posits that, for , where is the kth prime number, is the divisor function, and is an explicit error term that is negligible asymptotically; both the main term and error term represent enumerative functions in our conceptual model. We refine the error term to give numerical estimates of similar to those provided by the logarithmic integral, and much more accurate than up to where the estimates are almost exact. We then perform computational tests of unusual predictions of the model, finding limited evidence of predictable variations in prime gaps.
{"title":"Partition-theoretic model of prime distribution","authors":"Aidan Botkin , Madeline L. Dawsey , David J. Hemmer , Matthew R. Just , Robert Schneider","doi":"10.1016/j.jnt.2025.12.008","DOIUrl":"10.1016/j.jnt.2025.12.008","url":null,"abstract":"<div><div>We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that naturally predicts the prime number theorem as well as the twin prime conjecture. The model posits that, for <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>,<span><span><span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace></mspace><mo>=</mo><mspace></mspace><mn>1</mn><mspace></mspace><mo>+</mo><mspace></mspace><mn>2</mn><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mrow><mo>⌈</mo><mfrac><mrow><mi>d</mi><mo>(</mo><mi>j</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mspace></mspace><mo>+</mo><mspace></mspace><mi>ε</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is the <em>k</em>th prime number, <span><math><mi>d</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> is the divisor function, and <span><math><mi>ε</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> is an explicit error term that is negligible asymptotically; both the main term and error term represent enumerative functions in our conceptual model. We refine the error term to give numerical estimates of <span><math><mi>π</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> similar to those provided by the logarithmic integral, and much more accurate than <span><math><mi>li</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> up to <span><math><mi>n</mi><mo>=</mo><mn>10</mn><mo>,</mo><mn>000</mn></math></span> where the estimates are <em>almost exact</em>. We then perform computational tests of unusual predictions of the model, finding limited evidence of predictable variations in prime gaps.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 104-130"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190682","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}
Pub Date : 2026-06-01Epub Date: 2025-12-29DOI: 10.1016/j.jnt.2025.11.011
Naveen K. Godara , Renu Joshi , Eshita Mazumdar
This article focuses on the study of zero-sum invariants of finite non-abelian groups. We address two main problems: the first centers on the ordered Davenport constant and the second on Gao's constant. We establish a connection between the ordered Davenport constant and the small Davenport constant for a finite non-abelian group of even order, which in turn gives a relation with the Noether number. Additionally, we confirm a conjecture of Gao and Li for a non-abelian group of order , where p is a prime. Furthermore, we prove a conjecture that connects the ordered Davenport constant to the Loewy length for certain classes of finite 2-groups.
{"title":"Combinatorial invariants for certain classes of non-abelian groups","authors":"Naveen K. Godara , Renu Joshi , Eshita Mazumdar","doi":"10.1016/j.jnt.2025.11.011","DOIUrl":"10.1016/j.jnt.2025.11.011","url":null,"abstract":"<div><div>This article focuses on the study of zero-sum invariants of finite non-abelian groups. We address two main problems: the first centers on the ordered Davenport constant and the second on Gao's constant. We establish a connection between the ordered Davenport constant and the small Davenport constant for a finite non-abelian group of even order, which in turn gives a relation with the Noether number. Additionally, we confirm a conjecture of Gao and Li for a non-abelian group of order <span><math><mn>2</mn><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>, where <em>p</em> is a prime. Furthermore, we prove a conjecture that connects the ordered Davenport constant to the Loewy length for certain classes of finite 2-groups.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 44-63"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886043","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}
Pub Date : 2026-06-01Epub Date: 2025-12-29DOI: 10.1016/j.jnt.2025.11.010
Yeong-Wook Kwon , Subong Lim
In this paper, we investigate a Zagier duality between the Fourier coefficients of harmonic Maass–Jacobi–Poincaré series and those of weakly skew-holomorphic Jacobi–Poincaré series. We also verify a similar duality involving the skew-holomorphic Jacobi–Eisenstein series. As an application of these duality results, we show that the weakly skew-holomorphic Poincaré series and the skew-holomorphic Jacobi–Eisenstein series are orthogonal to the space of skew-holomorphic Jacobi cusp forms. Moreover, in the case of integral weight and level one, we obtain the rationality for the coefficients of the skew-holomorphic Jacobi–Eisenstein series. Combined with the duality result for the Jacobi–Eisenstein series, this implies the rationality of the constant term in the holomorphic part of the harmonic Maass–Jacobi–Poincaré series.
{"title":"Zagier duality for Jacobi forms","authors":"Yeong-Wook Kwon , Subong Lim","doi":"10.1016/j.jnt.2025.11.010","DOIUrl":"10.1016/j.jnt.2025.11.010","url":null,"abstract":"<div><div>In this paper, we investigate a Zagier duality between the Fourier coefficients of harmonic Maass–Jacobi–Poincaré series and those of weakly skew-holomorphic Jacobi–Poincaré series. We also verify a similar duality involving the skew-holomorphic Jacobi–Eisenstein series. As an application of these duality results, we show that the weakly skew-holomorphic Poincaré series and the skew-holomorphic Jacobi–Eisenstein series are orthogonal to the space of skew-holomorphic Jacobi cusp forms. Moreover, in the case of integral weight and level one, we obtain the rationality for the coefficients of the skew-holomorphic Jacobi–Eisenstein series. Combined with the duality result for the Jacobi–Eisenstein series, this implies the rationality of the constant term in the holomorphic part of the harmonic Maass–Jacobi–Poincaré series.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 15-43"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886044","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}
Pub Date : 2026-06-01Epub Date: 2025-12-29DOI: 10.1016/j.jnt.2025.11.014
Bikram Misra , M. Ram Murty , Biswajyoti Saha
We study the triple convolution sum of the divisor function given by for where denotes the number of positive divisors of n. Based on some algebraic and geometric considerations, Browning conjectured that the above sum is asymptotic to , for a suitable constant , as . This conjecture is still unproved. Using sieve-theoretic results of Wolke and Nair (respectively), it is possible to derive the exact order of the sum. The lower bound of the correct order of magnitude can also be derived by very elementary arguments. In this article, using the Tauberian theory for multiple Dirichlet series, we prove an explicit lower bound and provide a new theoretical framework to predict Browning's conjectured constant .
{"title":"A triple convolution sum of the divisor function","authors":"Bikram Misra , M. Ram Murty , Biswajyoti Saha","doi":"10.1016/j.jnt.2025.11.014","DOIUrl":"10.1016/j.jnt.2025.11.014","url":null,"abstract":"<div><div>We study the triple convolution sum of the divisor function given by<span><span><span><math><munder><mo>∑</mo><mrow><mi>n</mi><mo>≤</mo><mi>x</mi></mrow></munder><mi>d</mi><mo>(</mo><mi>n</mi><mo>)</mo><mi>d</mi><mo>(</mo><mi>n</mi><mo>−</mo><mi>h</mi><mo>)</mo><mi>d</mi><mo>(</mo><mi>n</mi><mo>+</mo><mi>h</mi><mo>)</mo></math></span></span></span> for <span><math><mi>h</mi><mo>≠</mo><mn>0</mn></math></span> where <span><math><mi>d</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denotes the number of positive divisors of <em>n</em>. Based on some algebraic and geometric considerations, Browning conjectured that the above sum is asymptotic to <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>h</mi></mrow></msub><mi>x</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>x</mi><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup></math></span>, for a suitable constant <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>≠</mo><mn>0</mn></math></span>, as <span><math><mi>x</mi><mo>→</mo><mo>∞</mo></math></span>. This conjecture is still unproved. Using sieve-theoretic results of Wolke and Nair (respectively), it is possible to derive the exact order of the sum. The lower bound of the correct order of magnitude can also be derived by very elementary arguments. In this article, using the Tauberian theory for multiple Dirichlet series, we prove an explicit lower bound and provide a new theoretical framework to predict Browning's conjectured constant <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 97-120"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927959","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}
Pub Date : 2026-06-01Epub Date: 2026-01-20DOI: 10.1016/j.jnt.2025.12.003
Ethan Simpson Lee , Paweł Nosal
In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that and are bounded from above by for all and respectively, where and are the Chebyshev ψ and ϑ functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld.
{"title":"Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis","authors":"Ethan Simpson Lee , Paweł Nosal","doi":"10.1016/j.jnt.2025.12.003","DOIUrl":"10.1016/j.jnt.2025.12.003","url":null,"abstract":"<div><div>In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that <span><math><mo>|</mo><mi>ψ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>x</mi><mo>|</mo></math></span> and <span><math><mo>|</mo><mi>ϑ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>x</mi><mo>|</mo></math></span> are bounded from above by<span><span><span><math><mfrac><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt><mi>log</mi><mo></mo><mi>x</mi><mo>(</mo><mi>log</mi><mo></mo><mi>x</mi><mo>−</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>x</mi><mo>)</mo></mrow><mrow><mn>8</mn><mi>π</mi></mrow></mfrac></math></span></span></span> for all <span><math><mi>x</mi><mo>≥</mo><mn>101</mn></math></span> and <span><math><mi>x</mi><mo>≥</mo><mn>2</mn><mspace></mspace><mn>657</mn></math></span> respectively, where <span><math><mi>ψ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><mi>ϑ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are the Chebyshev <em>ψ</em> and <em>ϑ</em> functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 241-258"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023475","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}
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