{"title":"On the values of Weierstrass zeta and sigma functions (with an appendix by David Masser)","authors":"K. Senthil Kumar","doi":"10.4064/aa230201-22-5","DOIUrl":"https://doi.org/10.4064/aa230201-22-5","url":null,"abstract":"","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70443124","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}
This article is composed of two parts. The first part concerns the general result on the variance associated with the distribution of a real sequence ${a_n}$ over arithmetic progressions, and the second part is an example of calculating the variance whe
{"title":"On a variance associated with the distribution of real sequences in arithmetic progressions","authors":"Pengyong Ding","doi":"10.4064/aa221003-14-8","DOIUrl":"https://doi.org/10.4064/aa221003-14-8","url":null,"abstract":"This article is composed of two parts. The first part concerns the general result on the variance associated with the distribution of a real sequence ${a_n}$ over arithmetic progressions, and the second part is an example of calculating the variance whe","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135260798","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 extend the usual Hilbert property for varieties over fields to arithmetic schemes over integral domains by demanding the set of near-integral points (as defined by Vojta) to be non-thin. We then generalize results of Bary-Soroker-Fehm-Petersen and Corvaja-Zannier by proving several structure results related to products and finite '{e}tale covers of arithmetic schemes with the Hilbert property.
{"title":"The Hilbert property for arithmetic schemes","authors":"Cedric Luger","doi":"10.4064/aa211214-16-11","DOIUrl":"https://doi.org/10.4064/aa211214-16-11","url":null,"abstract":"We extend the usual Hilbert property for varieties over fields to arithmetic schemes over integral domains by demanding the set of near-integral points (as defined by Vojta) to be non-thin. We then generalize results of Bary-Soroker-Fehm-Petersen and Corvaja-Zannier by proving several structure results related to products and finite '{e}tale covers of arithmetic schemes with the Hilbert property.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44816945","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}
Hildebrand proved that the smooth approximation for the number $Psi(x,y)$ of $y$-friable integers not exceeding $x$ holds for $y>(log x)^{2+varepsilon}$ under the Riemann hypothesis and conjectured that it fails when $yleqslant (log x)^{2-varepsilon}$. This conjecture has been recently confirmed by Gorodetsky by an intricate argument. We propose a short, straight-forward proof.
{"title":"Note on a conjecture of Hildebrand regarding friable integers","authors":"R. Bretèche, G. Tenenbaum","doi":"10.4064/aa221127-24-4","DOIUrl":"https://doi.org/10.4064/aa221127-24-4","url":null,"abstract":"Hildebrand proved that the smooth approximation for the number $Psi(x,y)$ of $y$-friable integers not exceeding $x$ holds for $y>(log x)^{2+varepsilon}$ under the Riemann hypothesis and conjectured that it fails when $yleqslant (log x)^{2-varepsilon}$. This conjecture has been recently confirmed by Gorodetsky by an intricate argument. We propose a short, straight-forward proof.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41507330","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 show that the cyclotomic conjecture on the characteristic polynomial of T-ramified S-split Iwasawa modules introduced in a previous paper and satisfied by abelian fields governs the Z${ell}$-rank of the submodule of fixed points for all finite disjoint sets S and T of places.Last, in the CM-case we prove that the weak and the strong versions of the cyclotomic conjecture both are equivalent to the conjunction of the classical conjectures of Leopoldt and Gross-Kuz'min.
{"title":"Conjecture cyclotomique et semi-simplicité des modules d’Iwasawa","authors":"J. Jaulent","doi":"10.4064/aa221123-27-4","DOIUrl":"https://doi.org/10.4064/aa221123-27-4","url":null,"abstract":"We show that the cyclotomic conjecture on the characteristic polynomial of T-ramified S-split Iwasawa modules introduced in a previous paper and satisfied by abelian fields governs the Z${ell}$-rank of the submodule of fixed points for all finite disjoint sets S and T of places.Last, in the CM-case we prove that the weak and the strong versions of the cyclotomic conjecture both are equivalent to the conjunction of the classical conjectures of Leopoldt and Gross-Kuz'min.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43342989","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 study the distribution of zeros of zeta functions associated to Beurling generalized prime number systems whose integers are distributed as $N(x) = Ax + O(x^{theta})$. We obtain in particular [ N(alpha, T) ll T^{frac{c(1-alpha)}{1-theta}}log^{9} T, ] for a constant $c$ arbitrarily close to $4$, improving significantly the current state of the art. We also investigate the consequences of the obtained zero-density estimates on the PNT in short intervals. Our proofs crucially rely on an extension of the classical mean-value theorem for Dirichlet polynomials to generalized Dirichlet polynomials.
{"title":"On zero-density estimates and the PNT in short intervals for Beurling generalized numbers","authors":"Frederik Broucke, Gregory Debruyne","doi":"10.4064/aa221223-15-2","DOIUrl":"https://doi.org/10.4064/aa221223-15-2","url":null,"abstract":"We study the distribution of zeros of zeta functions associated to Beurling generalized prime number systems whose integers are distributed as $N(x) = Ax + O(x^{theta})$. We obtain in particular [ N(alpha, T) ll T^{frac{c(1-alpha)}{1-theta}}log^{9} T, ] for a constant $c$ arbitrarily close to $4$, improving significantly the current state of the art. We also investigate the consequences of the obtained zero-density estimates on the PNT in short intervals. Our proofs crucially rely on an extension of the classical mean-value theorem for Dirichlet polynomials to generalized Dirichlet polynomials.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45086234","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}
Following Schmidt, Thurnheer and Bugeaud-Kristensen, we study how Dirichlet's theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make a bounded acute angle with respect to a fixed proper non-zero subspace $V$ of $mathbb{R}^n$. Assuming that the point of $mathbb{R}^n$ that we are approximating has linearly independent coordinates over $mathbb{Q}$, we obtain best possible exponents of approximation which surprisingly depend only on the dimension of $V$. Our estimates are derived by reduction to a result of Thurnheer, while their optimality follows from a new general construction in parametric geometry of numbers involving angular constraints.
{"title":"Diophantine approximation with constraints","authors":"J. Champagne, D. Roy","doi":"10.4064/aa221031-8-12","DOIUrl":"https://doi.org/10.4064/aa221031-8-12","url":null,"abstract":"Following Schmidt, Thurnheer and Bugeaud-Kristensen, we study how Dirichlet's theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make a bounded acute angle with respect to a fixed proper non-zero subspace $V$ of $mathbb{R}^n$. Assuming that the point of $mathbb{R}^n$ that we are approximating has linearly independent coordinates over $mathbb{Q}$, we obtain best possible exponents of approximation which surprisingly depend only on the dimension of $V$. Our estimates are derived by reduction to a result of Thurnheer, while their optimality follows from a new general construction in parametric geometry of numbers involving angular constraints.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47933610","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}
The main interest of this article is the one-sided boundedness of the local discrepancy of $alphainmathbb{R}setminusmathbb{Q}$ on the interval $(0,c)subset(0,1)$ defined by [D_n(alpha,c)=sum_{j=1}^n 1_{{{jalpha}
{"title":"On the one-sided boundedness of the local discrepancy of ${nalpha }$-sequences","authors":"J. Ying, Yushu Zheng","doi":"10.4064/aa211015-12-11","DOIUrl":"https://doi.org/10.4064/aa211015-12-11","url":null,"abstract":"The main interest of this article is the one-sided boundedness of the local discrepancy of $alphainmathbb{R}setminusmathbb{Q}$ on the interval $(0,c)subset(0,1)$ defined by [D_n(alpha,c)=sum_{j=1}^n 1_{{{jalpha}<c}}-cn.] We focus on the special case $cin (0,1)capmathbb{Q}$. Several necessary and sufficient conditions on $alpha$ for $(D_n(alpha,c))$ to be one-side bounded are derived. Using these, certain topological properties are given to describe the size of the set [O_c={alphain irr: (D_n(alpha,c)) text{ is one-side bounded}}.]","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41964348","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 give all possible holomorphic Eisenstein series on $Gamma_0(p)$, of rational weights greater than $2$, and with multiplier systems the same as certain rational-weight eta-quotients at all cusps. We prove they are modular forms and give their Fourier expansions. We establish four sorts of identities that equate such series to rational-weight eta-quotients. As an application, we give series expressions of special values of Euler Gamma function at any rational arguments. These expressions involve exponential sums of Dedekind sums.
{"title":"Holomorphic Eisenstein series of rational weights and special values of Gamma function","authors":"Xiaojie Zhu","doi":"10.4064/aa221110-1-4","DOIUrl":"https://doi.org/10.4064/aa221110-1-4","url":null,"abstract":"We give all possible holomorphic Eisenstein series on $Gamma_0(p)$, of rational weights greater than $2$, and with multiplier systems the same as certain rational-weight eta-quotients at all cusps. We prove they are modular forms and give their Fourier expansions. We establish four sorts of identities that equate such series to rational-weight eta-quotients. As an application, we give series expressions of special values of Euler Gamma function at any rational arguments. These expressions involve exponential sums of Dedekind sums.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47150248","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}
The volume of the unit sphere in every dimension is given a new interpretation as a product of special values of the zeta function of $mathbb{Z}$, akin to volume formulas of Minkowski and Siegel in the theory of arithmetic groups. A product formula is found for this zeta function that specializes to Catalan numbers. Moreover, certain closed-form expressions for various other zeta values are deduced, in particular leading to an alternative perspective on Euler's values of the Riemann zeta function.
{"title":"Volumes of spheres and special values\u0000of zeta functions of $mathbb{Z}$ and $mathbb{Z}/nmathbb{Z}$","authors":"A. Karlsson, Massimiliano Pallich","doi":"10.4064/aa220912-1-3","DOIUrl":"https://doi.org/10.4064/aa220912-1-3","url":null,"abstract":"The volume of the unit sphere in every dimension is given a new interpretation as a product of special values of the zeta function of $mathbb{Z}$, akin to volume formulas of Minkowski and Siegel in the theory of arithmetic groups. A product formula is found for this zeta function that specializes to Catalan numbers. Moreover, certain closed-form expressions for various other zeta values are deduced, in particular leading to an alternative perspective on Euler's values of the Riemann zeta function.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43651268","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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