In earlier work, we have shown that for certain geometric structures on a smooth manifold M of dimension n, one obtains an almost para-Kähler–Einstein metric on a manifold A of dimension 2n associated to the structure on M. The geometry also associates a diffeomorphism between A and $T^*M$ to any torsion-free connection compatible with the geometric structure. Hence we can use this construction to associate to each compatible connection an almost para-Kähler–Einstein metric on $T^*M$. In this short article, we discuss the relation of these metrics to Patterson–Walker metrics and derive explicit formulae for them in the cases of projective, conformal and Grassmannian structures.
在早先的工作中，我们已经证明，对于维数为 n 的光滑流形 M 上的某些几何结构，我们可以在维数为 2n 的流形 A 上得到一个与 M 上的结构相关联的近似对凯勒-爱因斯坦度量。因此，我们可以利用这一构造，在 $T^*M$ 上为每个兼容连接关联一个几乎准凯勒-爱因斯坦度量。在这篇短文中，我们将讨论这些度量与帕特森-沃克度量的关系，并推导出它们在投影结构、共形结构和格拉斯曼结构情况下的明确公式。
{"title":"Induced almost para-Kähler Einstein metrics on cotangent bundles","authors":"Andreas Čap, Thomas Mettler","doi":"10.1093/qmath/haae047","DOIUrl":"https://doi.org/10.1093/qmath/haae047","url":null,"abstract":"In earlier work, we have shown that for certain geometric structures on a smooth manifold M of dimension n, one obtains an almost para-Kähler–Einstein metric on a manifold A of dimension 2n associated to the structure on M. The geometry also associates a diffeomorphism between A and $T^*M$ to any torsion-free connection compatible with the geometric structure. Hence we can use this construction to associate to each compatible connection an almost para-Kähler–Einstein metric on $T^*M$. In this short article, we discuss the relation of these metrics to Patterson–Walker metrics and derive explicit formulae for them in the cases of projective, conformal and Grassmannian structures.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"12 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255739","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 study sumsets $mathcal{A}+mathcal{B}$ in the set of squares $mathcal{S}$ (and, more generally, in the set of kth powers $mathcal{S}_k$, where $kgeq 2$ is an integer). It is known by a result of Gyarmati that $mathcal{A}+mathcal{B}subset mathcal{S}_k cap [1,N]$ implies that $min(|mathcal{A}|,|mathcal{B}|)=O_k(log N)$. Here, we study how the upper bound on $|mathcal{B}|$ decreases, when the size of $|mathcal{A}|$ increases (or vice versa). In particular, if $|mathcal{A}|geq C k^{frac{1}{m}} m (log N)^{frac{1}{m}}$, then $|mathcal{B}|=O_k(m^2 log N)$, for sufficiently large N, a positive integer m and an explicit constant C > 0. For example, with $msim log log N$ this gives: If $|mathcal{A}|geq C_k log log N$, then $|mathcal{B}|=O_k(log N (log log N)^2)$.
我们研究平方集合 $mathcal{S}$ 中的和集 $mathcal{A}+mathcal{B}$ （更广义地说，是 kth 幂集合 $mathcal{S}_k$ 中的和集，其中 $kgeq 2$ 是整数）。由嘉尔马蒂的一个结果可知，$mathcal{A}+mathcal{B}subset mathcal{S}_k cap [1,N]$ 意味着 $min(|mathcal{A}|,|mathcal{B}|)=O_k(log N)$ 。在这里，我们将研究当 $||mathcal{A}|$ 的大小增加时，$||mathcal{B}|$ 的上界是如何减小的（反之亦然）。特别是，如果 $|mathcal{A}|geq C k^{frac{1}{m}} m (log N)^{frac{1}{m}}$, 那么 $|mathcal{B}|=O_k(m^2 log N)$, 对于足够大的 N、一个正整数 m 和一个显式常数 C > 0。 例如，在 $msim log log N$ 的情况下，这就给出了：如果$|mathcal{A}|geq C_k log log N$，那么$|mathcal{B}|=O_k(log N (log log N)^2)$。
{"title":"Sumsets in the set of squares","authors":"Christian Elsholtz, Lena Wurzinger","doi":"10.1093/qmath/haae044","DOIUrl":"https://doi.org/10.1093/qmath/haae044","url":null,"abstract":"We study sumsets $mathcal{A}+mathcal{B}$ in the set of squares $mathcal{S}$ (and, more generally, in the set of kth powers $mathcal{S}_k$, where $kgeq 2$ is an integer). It is known by a result of Gyarmati that $mathcal{A}+mathcal{B}subset mathcal{S}_k cap [1,N]$ implies that $min(|mathcal{A}|,|mathcal{B}|)=O_k(log N)$. Here, we study how the upper bound on $|mathcal{B}|$ decreases, when the size of $|mathcal{A}|$ increases (or vice versa). In particular, if $|mathcal{A}|geq C k^{frac{1}{m}} m (log N)^{frac{1}{m}}$, then $|mathcal{B}|=O_k(m^2 log N)$, for sufficiently large N, a positive integer m and an explicit constant C > 0. For example, with $msim log log N$ this gives: If $|mathcal{A}|geq C_k log log N$, then $|mathcal{B}|=O_k(log N (log log N)^2)$.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"84 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180087","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 compute some differentials of Sinha’s spectral sequence for cohomology of the space of long knots modulo immersions in codimension one, mainly over a field of characteristic 2 or 3. This spectral sequence is closely related to Vassiliev’s spectral sequence for the space of long knots in codimension $geq2$. We prove that the d2-differential of an element is non-zero in characteristic 2, which has already essentially been proved by Salvatore, and the d3-differential of another element is non-zero in characteristic 3. While the geometric meaning of the sequence is unclear in codimension one, these results have some applications to non-formality of operads. The result in characteristic 3 implies planar non-formality of the standard map $C_ast(E_1)to C_ast(E_2)$ in characteristic 3, where $C_ast(E_k)$ denotes the chain little k-disks operad. We also reprove the result of Salvatore which states that $C_ast(E_2)$ is not formal as a planar operad in characteristic 2 using the result in characteristic 2. For the computation, we transfer the structure on configuration spaces behind the spectral sequence onto Thom spaces over fat diagonals through a duality between configuration spaces and fat diagonals. This procedure enables us to describe the differentials by relatively simple maps to Thom spaces. We also show that the d2-differential of the generator of bidegree $(-4,2)$ is zero in characteristic $not=2$. This computation illustrates how one can manage the three-term relation using the description. Although the computations in this paper are concentrated to codimension one, our method also works for codimension $geq2$ and we prepare most of the basic notions and lemmas for general codimension.
{"title":"Sinha’s spectral sequence for long knots in codimension one and non-formality of the little 2-disks operad","authors":"Syunji Moriya","doi":"10.1093/qmath/haae043","DOIUrl":"https://doi.org/10.1093/qmath/haae043","url":null,"abstract":"We compute some differentials of Sinha’s spectral sequence for cohomology of the space of long knots modulo immersions in codimension one, mainly over a field of characteristic 2 or 3. This spectral sequence is closely related to Vassiliev’s spectral sequence for the space of long knots in codimension $geq2$. We prove that the d2-differential of an element is non-zero in characteristic 2, which has already essentially been proved by Salvatore, and the d3-differential of another element is non-zero in characteristic 3. While the geometric meaning of the sequence is unclear in codimension one, these results have some applications to non-formality of operads. The result in characteristic 3 implies planar non-formality of the standard map $C_ast(E_1)to C_ast(E_2)$ in characteristic 3, where $C_ast(E_k)$ denotes the chain little k-disks operad. We also reprove the result of Salvatore which states that $C_ast(E_2)$ is not formal as a planar operad in characteristic 2 using the result in characteristic 2. For the computation, we transfer the structure on configuration spaces behind the spectral sequence onto Thom spaces over fat diagonals through a duality between configuration spaces and fat diagonals. This procedure enables us to describe the differentials by relatively simple maps to Thom spaces. We also show that the d2-differential of the generator of bidegree $(-4,2)$ is zero in characteristic $not=2$. This computation illustrates how one can manage the three-term relation using the description. Although the computations in this paper are concentrated to codimension one, our method also works for codimension $geq2$ and we prepare most of the basic notions and lemmas for general codimension.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180090","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 study the codegree isomorphism problem for finite simple groups. In particular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters. The proof is uniform for all simple groups and only depends on the classification by means of Artin–Tits’ simple order theorem.
{"title":"The codegree isomorphism problem for finite simple groups","authors":"Nguyen N Hung, Alexander Moretó","doi":"10.1093/qmath/haae039","DOIUrl":"https://doi.org/10.1093/qmath/haae039","url":null,"abstract":"We study the codegree isomorphism problem for finite simple groups. In particular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters. The proof is uniform for all simple groups and only depends on the classification by means of Artin–Tits’ simple order theorem.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"2011 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141613702","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 study the homotopy groups of open books in terms of those of their pages and bindings. Under homotopy theoretic conditions on the monodromy we prove an integral decomposition result for the based loop space on an open book, and under more relaxed conditions we prove a rational loop space decomposition. The latter case allows for a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that for Milnor’s open book decomposition of an odd sphere with monodromy of finite order the induced action of the monodromy on the homology groups of its page cannot be nilpotent.
{"title":"Homotopy Theoretic Properties Of Open Books","authors":"Ruizhi Huang, Stephen Theriault","doi":"10.1093/qmath/haae035","DOIUrl":"https://doi.org/10.1093/qmath/haae035","url":null,"abstract":"We study the homotopy groups of open books in terms of those of their pages and bindings. Under homotopy theoretic conditions on the monodromy we prove an integral decomposition result for the based loop space on an open book, and under more relaxed conditions we prove a rational loop space decomposition. The latter case allows for a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that for Milnor’s open book decomposition of an odd sphere with monodromy of finite order the induced action of the monodromy on the homology groups of its page cannot be nilpotent.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"40 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141530597","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 the existence of non-commutative crepant resolutions (in the sense of Van den Bergh) of quotient singularities by finite and linearly reductive group schemes in positive characteristic. In dimension 2, we relate these to resolutions of singularities provided by G-Hilbert schemes and F-blowups. As an application, we establish and recover results concerning resolutions for toric singularities, as well as canonical, log terminal and F-regular singularities in dimension 2.
我们证明了正特征有限线性还原群方案的商奇点的非交换crepant决议（在Van den Bergh的意义上）的存在。在维度 2 中，我们把它们与 G-Hilbert 方案和 F-blowups 所提供的奇点解析联系起来。作为应用，我们建立并恢复了关于环奇点的解析结果，以及维 2 中的卡农、对数终端和 F 不规则奇点的解析结果。
{"title":"Non-commutative resolutions of linearly reductive quotient singularities","authors":"Christian Liedtke, Takehiko Yasuda","doi":"10.1093/qmath/haae033","DOIUrl":"https://doi.org/10.1093/qmath/haae033","url":null,"abstract":"We prove the existence of non-commutative crepant resolutions (in the sense of Van den Bergh) of quotient singularities by finite and linearly reductive group schemes in positive characteristic. In dimension 2, we relate these to resolutions of singularities provided by G-Hilbert schemes and F-blowups. As an application, we establish and recover results concerning resolutions for toric singularities, as well as canonical, log terminal and F-regular singularities in dimension 2.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501801","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 stronger form of our previous result that Schinzel’s Hypothesis holds for 100% of n-tuples of integer polynomials satisfying the usual necessary conditions, where the primes represented by the polynomials are subject to additional constraints in terms of Legendre symbols, as well as upper and lower bounds. We establish the triviality of the Brauer group of generic diagonal conic bundles over the projective line. Finally, we give an explicit lower bound for the probability that diagonal conic bundles in certain natural families have rational points.
我们证明了先前结果的更强形式，即对于满足通常必要条件的 n 组整数多项式，辛泽尔假说 100%成立，其中多项式所代表的素数受到 Legendre 符号以及上下限的额外约束。我们建立了投影线上一般对角圆锥束的布劳尔群的三性。最后，我们给出了某些自然系中对角圆锥束具有有理点的概率的明确下限。
{"title":"Generic Diagonal Conic Bundles Revisited","authors":"Alexei N Skorobogatov, Efthymios Sofos","doi":"10.1093/qmath/haae022","DOIUrl":"https://doi.org/10.1093/qmath/haae022","url":null,"abstract":"We prove a stronger form of our previous result that Schinzel’s Hypothesis holds for 100% of n-tuples of integer polynomials satisfying the usual necessary conditions, where the primes represented by the polynomials are subject to additional constraints in terms of Legendre symbols, as well as upper and lower bounds. We establish the triviality of the Brauer group of generic diagonal conic bundles over the projective line. Finally, we give an explicit lower bound for the probability that diagonal conic bundles in certain natural families have rational points.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"52 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501802","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 consider the subgroup of points of finite orbit through the action of an endomorphism of a finitely generated virtually free group, with particular emphasis on the subgroup of eventually fixed points, $text{EvFix}(varphi)$: points whose orbit contains a fixed point. We provide an algorithm to compute the subgroup of fixed points of an endomorphism of a finitely generated virtually free group and prove that finite orbits have cardinality bounded by a computable constant, which allows us to solve several algorithmic problems: deciding if φ is a finite order element of $text{End}(G)$, if φ is aperiodic, if $text{EvFix}(varphi)$ is finitely generated and if $text{EvFix}(varphi)$ is a normal subgroup. In the cases where $text{EvFix}(varphi)$ is finitely generated, we also present a bound for its rank and an algorithm to compute a generating set.
{"title":"Eventually fixed points of endomorphisms of virtually free groups","authors":"André Carvalho","doi":"10.1093/qmath/haae032","DOIUrl":"https://doi.org/10.1093/qmath/haae032","url":null,"abstract":"We consider the subgroup of points of finite orbit through the action of an endomorphism of a finitely generated virtually free group, with particular emphasis on the subgroup of eventually fixed points, $text{EvFix}(varphi)$: points whose orbit contains a fixed point. We provide an algorithm to compute the subgroup of fixed points of an endomorphism of a finitely generated virtually free group and prove that finite orbits have cardinality bounded by a computable constant, which allows us to solve several algorithmic problems: deciding if φ is a finite order element of $text{End}(G)$, if φ is aperiodic, if $text{EvFix}(varphi)$ is finitely generated and if $text{EvFix}(varphi)$ is a normal subgroup. In the cases where $text{EvFix}(varphi)$ is finitely generated, we also present a bound for its rank and an algorithm to compute a generating set.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"79 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501803","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}
Charles F Doran, Andrew Harder, Pierre Vanhove, Eric Pichon-Pharabod
We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into mixed Tate pieces and the Hodge structures of families of hyperelliptic, elliptic or rational curves depending on the space-time dimension. For two-loop graphs with a small number of edges, we give more precise results. In particular, we recover a result of Bloch (Double box motive. SIGMA 2021;17,048) that in the well-known double-box example, there is an underlying family of elliptic curves, and we give a concrete description of these elliptic curves. We show that the motive for the non-planar two-loop tardigrade graph is that of a K3 surface. In an appendix by Eric Pichon-Pharabod, we argue via high-precision numerical computations that the Picard number of this K3 surface is generically 11 and we compute the expected lattice polarization. Lastly, we show that generic members of the ice cream cone family of graph hypersurfaces correspond to the pairs of sunset Calabi–Yau varieties.
{"title":"Motivic Geometry of two-Loop Feynman Integrals","authors":"Charles F Doran, Andrew Harder, Pierre Vanhove, Eric Pichon-Pharabod","doi":"10.1093/qmath/haae015","DOIUrl":"https://doi.org/10.1093/qmath/haae015","url":null,"abstract":"We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into mixed Tate pieces and the Hodge structures of families of hyperelliptic, elliptic or rational curves depending on the space-time dimension. For two-loop graphs with a small number of edges, we give more precise results. In particular, we recover a result of Bloch (Double box motive. SIGMA 2021;17,048) that in the well-known double-box example, there is an underlying family of elliptic curves, and we give a concrete description of these elliptic curves. We show that the motive for the non-planar two-loop tardigrade graph is that of a K3 surface. In an appendix by Eric Pichon-Pharabod, we argue via high-precision numerical computations that the Picard number of this K3 surface is generically 11 and we compute the expected lattice polarization. Lastly, we show that generic members of the ice cream cone family of graph hypersurfaces correspond to the pairs of sunset Calabi–Yau varieties.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"18 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501804","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}
Elliott and Halberstam proved that $sum_{p lt n} 2^{omega(n-p)}$ is asymptotic to $phi(n)$. In analogy to the Erdős–Kac theorem, Elliott conjectured that if one restricts the summation to primes p such that $omega(n-p)le 2 log log n+lambda(2log log n)^{1/2}$ then the sum will be asymptotic to $phi(n)int_{-infty}^{lambda} mathrm{e}^{-t^2/2},mathrm{d}t/sqrt{2pi}$. We show that this conjecture follows from the Bombieri–Vinogradov theorem. We further prove a related result involving Poisson–Dirichlet distribution, employing deeper lying level of distribution results of the primes.
{"title":"On an ErdŐs–Kac-Type Conjecture of Elliott","authors":"Ofir Gorodetsky, Lasse Grimmelt","doi":"10.1093/qmath/haae026","DOIUrl":"https://doi.org/10.1093/qmath/haae026","url":null,"abstract":"Elliott and Halberstam proved that $sum_{p lt n} 2^{omega(n-p)}$ is asymptotic to $phi(n)$. In analogy to the Erdős–Kac theorem, Elliott conjectured that if one restricts the summation to primes p such that $omega(n-p)le 2 log log n+lambda(2log log n)^{1/2}$ then the sum will be asymptotic to $phi(n)int_{-infty}^{lambda} mathrm{e}^{-t^2/2},mathrm{d}t/sqrt{2pi}$. We show that this conjecture follows from the Bombieri–Vinogradov theorem. We further prove a related result involving Poisson–Dirichlet distribution, employing deeper lying level of distribution results of the primes.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"148 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191063","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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