The equation $x^2 + 1 = 0mod p$ has solutions whenever p = 2 or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. The roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec. The angles associated to the representation of such prime as a sum of squares are equidistributed is a famous theorem of Hecke. We give a natural way to associate between roots and angles and prove that the joint equidistribution of the sequence of pairs of roots and angles is equidistributed as well. Our approach involves an automorphic interpretation, which reduces the problem to the study of certain Poincare series on an arithmetic quotient of $SL_2(mathbb{R})$. Since our Poincare series have a nontrivial dependence on their Iwasawa θ-coordinate, they do not factor into functions on the upper half plane, as in the case studied by Duke et al. Spectral analysis on these higher dimensional varieties involves the nonspherical spectrum, making this paper the first complete study of a nonspherical equidistribution problem, with an arithmetic application. A couple of notable challenges we had to overcome were that of obtaining pointwise bounds for nonspherical Eisenstein series and utilizing a non-spherical analogue of the Selberg inversion formula, which we believe may have further implications beyond this work.
{"title":"Sectorial Equidistribution of the Roots of x2 + 1 Modulo Primes","authors":"Evgeny Musicantov, Sa’ar Zehavi","doi":"10.1093/qmath/haae011","DOIUrl":"https://doi.org/10.1093/qmath/haae011","url":null,"abstract":"The equation $x^2 + 1 = 0mod p$ has solutions whenever p = 2 or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. The roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec. The angles associated to the representation of such prime as a sum of squares are equidistributed is a famous theorem of Hecke. We give a natural way to associate between roots and angles and prove that the joint equidistribution of the sequence of pairs of roots and angles is equidistributed as well. Our approach involves an automorphic interpretation, which reduces the problem to the study of certain Poincare series on an arithmetic quotient of $SL_2(mathbb{R})$. Since our Poincare series have a nontrivial dependence on their Iwasawa θ-coordinate, they do not factor into functions on the upper half plane, as in the case studied by Duke et al. Spectral analysis on these higher dimensional varieties involves the nonspherical spectrum, making this paper the first complete study of a nonspherical equidistribution problem, with an arithmetic application. A couple of notable challenges we had to overcome were that of obtaining pointwise bounds for nonspherical Eisenstein series and utilizing a non-spherical analogue of the Selberg inversion formula, which we believe may have further implications beyond this work.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140322272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we develop foundational theory for geometries of the space of closed G2-structures in a given cohomology class as an infinite-dimensional manifold. We construct Levi-Civita connections for Sobolev-type metrics, formulate geodesic equations and analyze the variational structures of torsion-free G2-structures under these Sobolev-type metrics.
{"title":"The space of closed G2-structures. I. Connections","authors":"Pengfei Xu, Kai Zheng","doi":"10.1093/qmath/haae004","DOIUrl":"https://doi.org/10.1093/qmath/haae004","url":null,"abstract":"In this article, we develop foundational theory for geometries of the space of closed G2-structures in a given cohomology class as an infinite-dimensional manifold. We construct Levi-Civita connections for Sobolev-type metrics, formulate geodesic equations and analyze the variational structures of torsion-free G2-structures under these Sobolev-type metrics.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"160 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish the first explicit form of the Vinogradov–Korobov zero-free region for Dirichlet L-functions.
我们建立了德里赫特 L 函数的维诺格拉多夫-科罗波夫无零区域的第一种明确形式。
{"title":"An Explicit Vinogradov–Korobov Zero-Free Region for Dirichlet L-Functions","authors":"Tanmay Khale","doi":"10.1093/qmath/haae010","DOIUrl":"https://doi.org/10.1093/qmath/haae010","url":null,"abstract":"We establish the first explicit form of the Vinogradov–Korobov zero-free region for Dirichlet L-functions.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140167180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we analyze the convergence of the following type of series: $$ T_N ,,f(x)=sum_{j=N_1}^{N_2} v_jBig({mathcal P}_{a_{j+1}} ,,f(x)-{mathcal P}_{a_{j}} ,,f(x)Big),quad xin mathbb R_+, $$ where ${{mathcal P}_{t} }_{tgt0}$ is the Poisson semigroup associated with the Bessel operator $displaystyle Delta_lambda:=-{d^2over dx^2}-{2lambdaover x}{dover dx}$, with λ being a positive constant, $N=(N_1, N_2)in mathbb Z^2$ with $N_1 lt N_2,$ ${v_j}_{jin mathbb Z}$ is a bounded real sequence and ${a_j}_{jin mathbb Z}$ is an increasing real sequence. Our analysis will consist in the boundedness, in $L^p(mathbb{R}_+)$ and in $BMO(mathbb{R}_+)$, of the operators TN and its maximal operator $displaystyle T^*,,f(x)= sup_N leftvert T_N ,,f(x)rightvert.$ It is also shown that the local size of the maximal differential transform operators is the same with the order of a singular integral for functions f having local support.
{"title":"Boundedness of Differential Transforms for Poisson Semigroups Generated by Bessel Operators","authors":"Chao Zhang","doi":"10.1093/qmath/haae009","DOIUrl":"https://doi.org/10.1093/qmath/haae009","url":null,"abstract":"In this paper we analyze the convergence of the following type of series: $$ T_N ,,f(x)=sum_{j=N_1}^{N_2} v_jBig({mathcal P}_{a_{j+1}} ,,f(x)-{mathcal P}_{a_{j}} ,,f(x)Big),quad xin mathbb R_+, $$ where ${{mathcal P}_{t} }_{tgt0}$ is the Poisson semigroup associated with the Bessel operator $displaystyle Delta_lambda:=-{d^2over dx^2}-{2lambdaover x}{dover dx}$, with λ being a positive constant, $N=(N_1, N_2)in mathbb Z^2$ with $N_1 lt N_2,$ ${v_j}_{jin mathbb Z}$ is a bounded real sequence and ${a_j}_{jin mathbb Z}$ is an increasing real sequence. Our analysis will consist in the boundedness, in $L^p(mathbb{R}_+)$ and in $BMO(mathbb{R}_+)$, of the operators TN and its maximal operator $displaystyle T^*,,f(x)= sup_N leftvert T_N ,,f(x)rightvert.$ It is also shown that the local size of the maximal differential transform operators is the same with the order of a singular integral for functions f having local support.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"63 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140167148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a universality theorem for the Selberg zeta function of subgroups of $mathrm{SL}_2(mathbb{Z})$ or co-compact arithmetic groups derived from quaternion algebras, in the strip ${5/6 lt mathrm{Re}{s} lt 1}$, improving the range compared with a previous work by Drungilas–Garunkštis–Kačenas. We also obtain the same range for a joint universality theorem for congruence subgroups, which improves a result by Mishou.
{"title":"Universality theorems of Selberg zeta functions for arithmetic groups","authors":"Yasufumi Hashimoto","doi":"10.1093/qmath/haae006","DOIUrl":"https://doi.org/10.1093/qmath/haae006","url":null,"abstract":"We prove a universality theorem for the Selberg zeta function of subgroups of $mathrm{SL}_2(mathbb{Z})$ or co-compact arithmetic groups derived from quaternion algebras, in the strip ${5/6 lt mathrm{Re}{s} lt 1}$, improving the range compared with a previous work by Drungilas–Garunkštis–Kačenas. We also obtain the same range for a joint universality theorem for congruence subgroups, which improves a result by Mishou.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"3 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140003920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we prove an exact formula for the number of partitions without sequences. By work of Andrews, the corresponding generating function is a mixed mock modular form weight of 0. The proof requires evaluating and bounding Kloosterman sums and the Circle Method.
{"title":"A Rademacher-type exact formula for partitions without sequences","authors":"Walter Bridges, Kathrin Bringmann","doi":"10.1093/qmath/haad043","DOIUrl":"https://doi.org/10.1093/qmath/haad043","url":null,"abstract":"In this paper, we prove an exact formula for the number of partitions without sequences. By work of Andrews, the corresponding generating function is a mixed mock modular form weight of 0. The proof requires evaluating and bounding Kloosterman sums and the Circle Method.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"56 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140004069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that for large integers n, whose ratios of consecutive divisors are bound above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C log_2 n$ and variance $V log_2 n$, where $C=1/(1-{rm e}^{-gamma})approx 2.280$ and V ≈ 0.414. This result is then generalized in two different directions.
{"title":"An ErdŐs–Kac theorem for integers with dense divisors","authors":"Gérald Tenenbaum, Andreas Weingartner","doi":"10.1093/qmath/haae002","DOIUrl":"https://doi.org/10.1093/qmath/haae002","url":null,"abstract":"We show that for large integers n, whose ratios of consecutive divisors are bound above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C log_2 n$ and variance $V log_2 n$, where $C=1/(1-{rm e}^{-gamma})approx 2.280$ and V ≈ 0.414. This result is then generalized in two different directions.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139947682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, a mode of convergence for measurable functions is introduced. A related notion of Cauchy sequence is given, and it is proved that this notion of convergence is complete in the sense that Cauchy sequences converge. Moreover, the preservation of convergence under composition is investigated. The origin of this mode of convergence lies in the path of proving that the density of a Euler system converges almost everywhere (up to subsequences) towards the density of a non-linear diffusion system, as a consequence of the convergence in the relaxation limit.
{"title":"A mode of convergence arising in diffusive relaxation","authors":"Nuno J Alves, João Paulos","doi":"10.1093/qmath/haae001","DOIUrl":"https://doi.org/10.1093/qmath/haae001","url":null,"abstract":"In this work, a mode of convergence for measurable functions is introduced. A related notion of Cauchy sequence is given, and it is proved that this notion of convergence is complete in the sense that Cauchy sequences converge. Moreover, the preservation of convergence under composition is investigated. The origin of this mode of convergence lies in the path of proving that the density of a Euler system converges almost everywhere (up to subsequences) towards the density of a non-linear diffusion system, as a consequence of the convergence in the relaxation limit.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139947680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given an arbitrary, finitely presented, residually finite group Γ, one can construct a finitely generated, residually finite, free-by-free group $M_Gamma = F_inftyrtimes F_4$ and an embedding $M_Gamma hookrightarrow (F_4ast Gamma)times F_4$ that induces an isomorphism of profinite completions. In particular, there is a free-by-free group whose profinite completion contains $widehat{Gamma}$ as a retract.
{"title":"Profinite completions of free-by-free groups contain everything","authors":"Martin R Bridson","doi":"10.1093/qmath/haae003","DOIUrl":"https://doi.org/10.1093/qmath/haae003","url":null,"abstract":"Given an arbitrary, finitely presented, residually finite group Γ, one can construct a finitely generated, residually finite, free-by-free group $M_Gamma = F_inftyrtimes F_4$ and an embedding $M_Gamma hookrightarrow (F_4ast Gamma)times F_4$ that induces an isomorphism of profinite completions. In particular, there is a free-by-free group whose profinite completion contains $widehat{Gamma}$ as a retract.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"49 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139903385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $dinmathbb{N}$ and π be a fixed cuspidal automorphic representation of $mathrm{GL}_{d}(mathbb{A}_{mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values $-frac{L^{prime}}{L}(1+it,piotimeschi_D)$ as D varies over fundamental discriminants. Here, t is a fixed real number and χD is the real character associated with D. We establish an upper bound on the discrepancy in the convergence of this family to its limiting distribution. As an application of this result, we obtain an upper bound on the small values of $left|frac{L^{prime}}{L}(1,piotimeschi_D)right|$ when π is self-dual.
让 $dinmathbb{N}$ 和 π 是 $mathrm{GL}_{d}(mathbb{A}_{mathbb{Q}})$ 的一个具有单元中心特征的固定的尖顶自定形表示。我们确定了当 D 随基本判别式变化时，$-frac{L^{prime}}{L}(1+it,piotimeschi_D)$ 的值族的极限分布。这里，t 是一个固定实数，χD 是与 D 相关的实数特征。我们建立了这个族收敛到其极限分布的差异上限。作为这一结果的应用，我们得到了当π是自偶数时$left|frac{L^{prime}}{L}(1,piotimeschi_D)right|$的小值的上界。
{"title":"Value Distribution of Logarithmic Derivatives of Quadratic Twists of Automorphic L-functions","authors":"Amir Akbary, Alia Hamieh","doi":"10.1093/qmath/haad042","DOIUrl":"https://doi.org/10.1093/qmath/haad042","url":null,"abstract":"Let $dinmathbb{N}$ and π be a fixed cuspidal automorphic representation of $mathrm{GL}_{d}(mathbb{A}_{mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values $-frac{L^{prime}}{L}(1+it,piotimeschi_D)$ as D varies over fundamental discriminants. Here, t is a fixed real number and χD is the real character associated with D. We establish an upper bound on the discrepancy in the convergence of this family to its limiting distribution. As an application of this result, we obtain an upper bound on the small values of $left|frac{L^{prime}}{L}(1,piotimeschi_D)right|$ when π is self-dual.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"4 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139483557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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