{"results":[{"id":"ss_2cada177d1d4f5e03b1a7268c6d13ba9a9820d40","title":"Zeta Zeros in a Narrow Vertical Box","authors":[{"name":"D. Goldston"},{"name":"Ade Irma Suriajaya"}],"abstract":"In 1973 Montgomery proved, assuming the Riemann Hypothesis (RH), that asymptotically at least 2/3 of zeros of the Riemann zeta-function are simple zeros. In a previous note (arXiv:2511.20059 [math.NT]) we showed how RH can be replaced with a general estimate for a double sum over zeros, and this allows one to then obtain results on zeros that are both simple and on the critical line. Here we give a simple proof based on a direct generalization of Montgomery's proof that on assuming all the zeros are in a narrow vertical box between height $T$ and $2T$ of width $b/\\log T$ and centered on the critical line, then, if $b=b(T)\\to 0$ as $T\\to \\infty$, we have asymptotically at least 2/3 of the zeros are simple and on the critical line.","source":"Semantic Scholar","year":2026,"language":"en","subjects":["Mathematics"],"url":"https://www.semanticscholar.org/paper/2cada177d1d4f5e03b1a7268c6d13ba9a9820d40","is_open_access":true,"published_at":"","score":70},{"id":"ss_08d6433c9f72c30c91b51ed120d45722135e6d72","title":"Classification of Finite Groups With Equal Left and Right Quotient Sets","authors":[{"name":"Haran Mouli"},{"name":"Pramana Saldin"}],"abstract":"In this paper, we classify all finite groups $G$ which have the following property: for all subsets $A \\subseteq G$, we have $|AA^{-1}| = |A^{-1}A|$. This question is motivated by the problem in additive combinatorics of More Sums Than Difference sets and answers several questions posed in arXiv:2509.00611 [math.NT].","source":"Semantic Scholar","year":2025,"language":"en","subjects":["Mathematics"],"url":"https://www.semanticscholar.org/paper/08d6433c9f72c30c91b51ed120d45722135e6d72","is_open_access":true,"citations":1,"published_at":"","score":69.03},{"id":"ss_ce10ed7fcdd4abe01dfd91155f239177454bea83","title":"Vanishing properties of Kloosterman sums and Dyson’s conjectures","authors":[{"name":"Qihang Sun"}],"abstract":"In a previous paper (Sun, in: arXiv:2406.06294 [math.NT], 2024), the author proved the exact formulae for ranks of partitions modulo each prime p≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 5$$\\end{document}. In this paper, for p=5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=5$$\\end{document} and 7, we prove special vanishing properties of the Kloosterman sums appearing in the exact formulae. These vanishing properties imply a new proof of Dyson’s rank conjectures. Specifically, we give a new proof of Ramanujan’s congruences p(5n+4)≡0(mod5)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(5n+4)\\equiv 0\\ (\\textrm{mod}\\ 5)$$\\end{document} and p(7n+5)≡0(mod7)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(7n+5)\\equiv 0\\ (\\textrm{mod}\\ 7)$$\\end{document}.","source":"Semantic Scholar","year":2024,"language":"en","subjects":["Mathematics"],"doi":"10.1007/s11139-024-01011-4","url":"https://www.semanticscholar.org/paper/ce10ed7fcdd4abe01dfd91155f239177454bea83","is_open_access":true,"citations":2,"published_at":"","score":68.06},{"id":"ss_a40e387afcb88db97b7e5d276cf61a6e72b201af","title":"Generic decompositions of Deligne--Lusztig representations","authors":[{"name":"Daniel Le"},{"name":"B. Hung"},{"name":"B. Levin"},{"name":"Stefano Morra"}],"abstract":"Let $G_0$ be a reductive group over $\\mathbb{F}_p$ with simply connected derived subgroup, (geometrically) connected center and Coxeter number $h+1$. We extend Jantzen's generic decomposition pattern from $(2h-1)$-generic to $h$-generic Deligne--Lusztig representations, which is optimal. We also prove several results on the ``obvious'' Jordan--H\\\"older factors of general Deligne--Lusztig representations. As an application we improve the weight elimination result of arXiv:1610.04819 [math.NT]","source":"Semantic Scholar","year":2024,"language":"en","subjects":["Mathematics"],"url":"https://www.semanticscholar.org/paper/a40e387afcb88db97b7e5d276cf61a6e72b201af","is_open_access":true,"citations":1,"published_at":"","score":68.03},{"id":"arxiv_2410.09357","title":"Comment on \"On the squarefree density of polynomials\"","authors":[{"name":"Yuri G. Zarhin"}],"abstract":"This is an exposition of results of R.C. Vaughan and the author (Mathematika 70 (2024), no. 4). We discuss how often the squarefree values of an integral polynomial do occur. We discuss interrelations between our results and results of B. Poonen (arXiv:math/0203292 [math.NT]).","source":"arXiv","year":2024,"language":"en","subjects":["math.NT"],"url":"https://arxiv.org/abs/2410.09357","pdf_url":"https://arxiv.org/pdf/2410.09357","is_open_access":true,"published_at":"2024-10-12T03:52:13Z","score":68},{"id":"arxiv_2406.06294","title":"Exact formulae for ranks of partitions","authors":[{"name":"Qihang Sun"}],"abstract":"In 2009, Bringmann arXiv:0708.0691 [math.NT] used the circle method to prove an asymptotic formula for the Fourier coefficients of rank generating functions. In this paper, we prove that Bringmann's formula, when summing up to infinity and in the case of prime modulus, gives a Rademacher-type exact formula involving sums of vector-valued Kloosterman sums. As a corollary, in another paper arXiv:2406.07469 [math.NT], we will provide a new proof of Dyson's conjectures by showing that the certain Kloosterman sums vanish.","source":"arXiv","year":2024,"language":"en","subjects":["math.NT"],"doi":"10.1090/tran/9539","url":"https://arxiv.org/abs/2406.06294","pdf_url":"https://arxiv.org/pdf/2406.06294","is_open_access":true,"published_at":"2024-06-10T14:15:47Z","score":68},{"id":"arxiv_2409.03413","title":"New bound on small range sum polynomials of degree","authors":[{"name":"Ádám Markó"}],"abstract":"The polynomials of degree $\\frac{p-1}{2}$ of range sum $p$ was determined in {\\tt arXiv:2311.06136 [math.NT]} for large enough primes. We extend this result by reducing the lower bound for the primes to $23$ by introducing a new and elementary way of estimating sums of Legendre symbols.","source":"arXiv","year":2024,"language":"en","subjects":["math.NT"],"url":"https://arxiv.org/abs/2409.03413","pdf_url":"https://arxiv.org/pdf/2409.03413","is_open_access":true,"published_at":"2024-09-05T11:01:56Z","score":68},{"id":"ss_cda25a29cda74d4d78af76ea344824a4e8a2b58b","title":"$p$-adic properties of Eisenstein-Kronecker cocycles over imaginary quadratic fields and $p$-adic interpolation","authors":[{"name":"Jorge Fl'orez"}],"abstract":"We establish integrality and congruence properties for the Eisenstein-Kronecker cocycle of Bergeron, Charollois and Garc\\'ia introduced in [arXiv:2107.01992v2 [math.NT]]. As a consequence, we recover the integrality of the critical values of Hecke $L$-functions over imaginary quadratic fields in the split case. Additionally, we construct a $p$-adic measure that interpolates these critical values.","source":"Semantic Scholar","year":2024,"language":"en","subjects":["Mathematics"],"url":"https://www.semanticscholar.org/paper/cda25a29cda74d4d78af76ea344824a4e8a2b58b","is_open_access":true,"published_at":"","score":68},{"id":"ss_4cf0118fb67c3162aa9a12b3b15f86afc5e6ce9e","title":"On the order of magnitude of certain integer sequences","authors":[{"name":"M. Hellus"},{"name":"A. Rechenauer"},{"name":"R. Waldi"}],"abstract":"Let $p$ be a prime number, and let $S$ be the numerical semigroup generated by the prime numbers not less than $p$. We compare the orders of magnitude of some invariants of $S$ with each other, e. g., the biggest atom $u$ of $S$ with $p$ itself: By Harald Helfgott (arXiv:1312.7748 [math.NT]), every odd integer $N$ greater than five can be written as the sum of three prime numbers. There is numerical evidence suggesting that the summands of $N$ always can be chosen between $\\frac N6$ and $\\frac N2$. This would imply that $u$ is less than $6p$.","source":"Semantic Scholar","year":2024,"language":"en","subjects":["Mathematics"],"url":"https://www.semanticscholar.org/paper/4cf0118fb67c3162aa9a12b3b15f86afc5e6ce9e","is_open_access":true,"published_at":"","score":68},{"id":"ss_03502d2979ddcdeed4cbefdcc4a30582ded32b0b","title":"Large Sums of Fourier Coefficients of Cusp Forms","authors":[{"name":"Claire Fréchette"},{"name":"Mathilde Gerbelli-Gauthier"},{"name":"Ali Hamieh"},{"name":"Naomi Tanabe"}],"abstract":"Let $N$ be a fixed positive integer, and let $f\\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\\sum_{n=1}^{\\infty} \\lambda_f(n)n^{\\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\\sum_{n\\leq x}\\lambda_f(x)$. It is conjectured that $S(x,f)=o(x\\log x)$ in the range $x\\geq k^{\\epsilon}$. Lamzouri proved in arXiv:1703.10582 [math.NT] that this is true under the assumption of the Generalized Riemann Hypothesis (GRH) for $L(s,f)$. In this paper, we prove that this conjecture holds under a weaker assumption than GRH. In particular, we prove that given $\\epsilon\u003e(\\log k)^{-\\frac{1}{8}}$ and $1\\leq T\\leq (\\log k)^{\\frac{1}{200}}$, we have $S(x,f)\\ll \\frac{x\\log x}{T}$ in the range $x\\geq k^{\\epsilon}$ provided that $L(s,f)$ has no more than $\\epsilon^2\\log k/5000$ zeros in the region $\\left\\{s\\,:\\, \\Re(s)\\geq \\frac34, \\, |\\Im(s)-\\phi| \\leq \\frac14\\right\\}$ for every real number $\\phi$ with $|\\phi|\\leq T$.","source":"Semantic Scholar","year":2023,"language":"en","subjects":["Mathematics"],"url":"https://www.semanticscholar.org/paper/03502d2979ddcdeed4cbefdcc4a30582ded32b0b","is_open_access":true,"citations":3,"published_at":"","score":67.09},{"id":"ss_4ce656e09ef3935448ecead735dce8123ab9abff","title":"Dynamics on ℙ1: preperiodic points and pairwise stability","authors":[{"name":"Laura Demarco"},{"name":"N. Mavraki"}],"abstract":"DeMarco, Krieger, and Ye conjectured that there is a uniform bound B, depending only on the degree d, so that any pair of holomorphic maps $f, g :{\\mathbb {P}}^1\\to {\\mathbb {P}}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\\mathrm {Rat}_d \\times \\mathrm {Rat}_d$, for each degree $d\\geq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949–1001; Common preperiodic points for quadratic polynomials, J. Mod. Dyn. 18 (2022), 363–413] and of Poineau [Dynamique analytique sur $\\mathbb {Z}$ II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic curves.","source":"Semantic Scholar","year":2022,"language":"en","subjects":["Mathematics"],"doi":"10.1112/S0010437X23007546","url":"https://www.semanticscholar.org/paper/4ce656e09ef3935448ecead735dce8123ab9abff","is_open_access":true,"citations":14,"published_at":"","score":66.42},{"id":"ss_2c64b43d01fc95564602d3e2dbf83a067e163dbd","title":"Applications of analytic newvectors for GL(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm {GL}(n)$$\\end{doc","authors":[{"name":"S. Jana"}],"abstract":"We provide a few natural applications of the analytic newvectors, initiated in Jana and Nelson (Analytic newvectors for GLn(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\text {GL}_n(\\mathbb {R})$$\\end{document}, arXiv:1911.01880 [math.NT], 2019), to some analytic questions in automorphic forms for PGLn(Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm {PGL}_n(\\mathbb {Z})$$\\end{document} with n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 2$$\\end{document}, in the archimedean analytic conductor aspect. We prove an orthogonality result of the Fourier coefficients, a density estimate of the non-tempered forms, an equidistribution result of the Satake parameters with respect to the Sato–Tate measure, and a second moment estimate of the central L-values as strong as Lindelöf on average. We also prove the random matrix prediction about the distribution of the low-lying zeros of automorphic L-function in the analytic conductor aspect. The new ideas of the proofs include the use of analytic newvectors to construct an approximate projector on the automorphic spectrum with bounded conductors and a soft local (both at finite and infinite places) analysis of the geometric side of the Kuznetsov trace formula.","source":"Semantic Scholar","year":2021,"language":"en","subjects":null,"doi":"10.1007/s00208-021-02207-5","url":"https://www.semanticscholar.org/paper/2c64b43d01fc95564602d3e2dbf83a067e163dbd","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00208-021-02207-5.pdf","is_open_access":true,"citations":10,"published_at":"","score":65.3},{"id":"ss_e850350f83d730834d8425d46e9cefd394dc2f9e","title":"Homotopy Spectra and Diophantine Equations","authors":[{"name":"Y. Manin"},{"name":"M. Marcolli"}],"abstract":"Arguably, the first bridge between vast, ancient, but disjoint domains of mathematical knowledge, – topology and number theory, – was built only during the last fifty years. This bridge is the theory of spectra in the stable homotopy theory. In particular, it connects Z, the initial object in the theory of commutative rings, with the sphere spectrum S: see [Sc01] for one of versions of it. This connection poses the challenge: discover a new information in number theory using the developed independently machinery of homotopy theory. (Notice that a passage in reverse direction has already generated results about computability in the homotopy theory: see [FMa20] and references therein.) In this combined research/survey paper we suggest to apply homotopy spectra to the problem of distribution of rational points upon algebraic manifolds. Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT); Topology (math.AT) Comments: 72 pages. MSC-classes: 16E35, 11G50, 14G40, 55P43, 16E20, 18F30. CONTENTS 0. Introduction and summary 1. Homotopy spectra: a brief presentation 2. Diophantine equations: distribution of rational points on algebraic varieties 3. Rational points, sieves, and assemblers 4. Anticanonical heights and points count 5. Sieves “beyond heights” ? 6. Obstructions and sieves 7. Assemblers and spectra for Grothendieck rings with exponentials References 1","source":"Semantic Scholar","year":2021,"language":"en","subjects":["Mathematics"],"url":"https://www.semanticscholar.org/paper/e850350f83d730834d8425d46e9cefd394dc2f9e","is_open_access":true,"citations":1,"published_at":"","score":65.03},{"id":"arxiv_2111.03548","title":"Spectrum of p-adic linear differential equations I: The shape of the spectrum","authors":[{"name":"Tinhinane A. Azzouz"}],"abstract":"This paper extends our previous works arXiv:1802.07306 [math.NT], arXiv:1808.02382 [math.NT] on determining the spectrum, in the Berkovich sense, of ultrametric linear differential equations. Our previous works focused on equations with constant coefficients or over a field of formal power series. In this paper, we investigate the spectrum of $p$-adic differential equations at a generic point on a quasi-smooth curve. This analysis allows us to establish a significant connection between the spectrum and the spectral radii of convergence of a differential equation when considering the affine line. Furthermore, the spectrum offers a more detailed decomposition compared to Robba's decomposition based on spectral radii.","source":"arXiv","year":2021,"language":"en","subjects":["math.NT","math.SP"],"doi":"10.1007/s00029-023-00904-4","url":"https://arxiv.org/abs/2111.03548","pdf_url":"https://arxiv.org/pdf/2111.03548","is_open_access":true,"published_at":"2021-11-05T15:13:21Z","score":65},{"id":"doaj_10.46298/epiga.2019.volume3.4792","title":"Smooth affine group schemes over the dual numbers","authors":[{"name":"Matthieu ROMAGNY"},{"name":"Dajano Tossici"}],"abstract":"We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 \\to \\text{Lie}(G, I) \\to E \\to G \\to 1$ where G is an affine, smooth group scheme over k. Here k is an arbitrary commutative ring and $k[I] = k \\oplus I$ with $I^2 = 0$. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. It is compatible with the exact structures and the $\\mathbb{O}_k$-module stack structures on both categories. Our constructions rely on the use of the group algebra scheme of an affine group scheme; we introduce this object and establish its main properties. As an application, we establish a Dieudonné classification for smooth, commutative, unipotent group schemes over $k[I]$.","source":"DOAJ","year":2019,"language":"","subjects":["Mathematics"],"doi":"10.46298/epiga.2019.volume3.4792","url":"https://epiga.episciences.org/4792/pdf","pdf_url":"https://epiga.episciences.org/4792/pdf","is_open_access":true,"published_at":"","score":63},{"id":"ss_73eef61fc858f71564db0ddb1ac6e490afec72b2","title":"The divisibility by 2 of rational points on elliptic curves","authors":[{"name":"B. Bekker"},{"name":"Y. Zarhin"}],"abstract":"We give a simple proof of the well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion. As an application of the explicit division by $2^n$ formulas obtained in Sec.2, we construct versal families of elliptic curves containing points of orders 4, 5, 6, and 8 from which we obtain an explicit description of elliptic curves over certain finite fields $\\mathbb{F}_q$ with a prescribed (small) group $E(\\mathbb{F}_q)$. In the last two sections we study 3- and 5-torsion. This paper supercedes arXiv:1605.09279 [math.NT] .","source":"Semantic Scholar","year":2017,"language":"en","subjects":["Mathematics"],"url":"https://www.semanticscholar.org/paper/73eef61fc858f71564db0ddb1ac6e490afec72b2","is_open_access":true,"citations":9,"published_at":"","score":61.27},{"id":"doaj_10.23638/DMTCS-19-3-13","title":"Periodic balanced binary triangles","authors":[{"name":"Jonathan Chappelon"}],"abstract":"A binary triangle of size $n$ is a triangle of zeroes and ones, with $n$ rows, built with the same local rule as the standard Pascal triangle modulo $2$. A binary triangle is said to be balanced if the absolute difference between the numbers of zeroes and ones that constitute this triangle is at most $1$. In this paper, the existence of balanced binary triangles of size $n$, for all positive integers $n$, is shown. This is achieved by considering periodic balanced binary triangles, that are balanced binary triangles where each row, column or diagonal is a periodic sequence.","source":"DOAJ","year":2017,"language":"","subjects":["Mathematics"],"doi":"10.23638/DMTCS-19-3-13","url":"https://dmtcs.episciences.org/3141/pdf","pdf_url":"https://dmtcs.episciences.org/3141/pdf","is_open_access":true,"published_at":"","score":61},{"id":"arxiv_1712.05047","title":"Families of elliptic curves with rational torsion points of even order","authors":[{"name":"Boris M. Bekker"},{"name":"Yuri G. Zarhin"}],"abstract":"This paper is a follow up of arXiv:1702.02255 [math.NT]. We construct explicitly versal families of elliptic curves with rational points of order 4, 6, 8, 10, 12 respectively.","source":"arXiv","year":2017,"language":"en","subjects":["math.AG","math.NT"],"url":"https://arxiv.org/abs/1712.05047","pdf_url":"https://arxiv.org/pdf/1712.05047","is_open_access":true,"published_at":"2017-12-13T23:29:53Z","score":61}],"total":1082437,"page":1,"page_size":20,"sources":["DOAJ","arXiv","Semantic Scholar","CrossRef"],"query":"math.NT"}