{"title":"Abstract intersection theory for zeta-functions: geometric aspects","authors":"Grzegorz Banaszak, Y. Uetake","doi":"10.7169/FACM/1916","DOIUrl":"https://doi.org/10.7169/FACM/1916","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48980004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let k be a number field and let H(k) denote the Hilbert class field of k, that is the maximal abelian unramified extension of k. It is known by class field theory that the Galois group of the extension H(k)/k, i.e., G := Gal(H(k)/k), is isomorphic to Cl(k), the class group of k (cf. [13, p. 228]). The Hilbert genus field of k, denoted by E(k), is the invariant field of G. Thus, by Galois theory, we have: Cl(k)/Cl(k) ≃ G/G ≃ Gal(E(k)/k),
设k为数域，H(k)表示k的Hilbert类域，即k的最大阿贝尔无分支扩展。由类场论可知，扩展H(k)/k的伽罗瓦群，即G:= Gal(H(k)/k)，与k的类群Cl(k)同构(cf. [13, p. 228])。k的Hilbert格场，用E(k)表示，是G的不变场。因此，根据伽罗瓦理论，我们得到:Cl(k)/Cl(k)≃G/G≃Gal(E(k)/k);
{"title":"The construction of the Hilbert genus fields of real cyclic quartic fields","authors":"M. M. Chems-Eddin, Moulay Ahmed Hajjami, M. Taous","doi":"10.7169/facm/2014","DOIUrl":"https://doi.org/10.7169/facm/2014","url":null,"abstract":"Let k be a number field and let H(k) denote the Hilbert class field of k, that is the maximal abelian unramified extension of k. It is known by class field theory that the Galois group of the extension H(k)/k, i.e., G := Gal(H(k)/k), is isomorphic to Cl(k), the class group of k (cf. [13, p. 228]). The Hilbert genus field of k, denoted by E(k), is the invariant field of G. Thus, by Galois theory, we have: Cl(k)/Cl(k) ≃ G/G ≃ Gal(E(k)/k),","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44343428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we establish quantitative results for sign changes in certain subsequences of primitive Fourier coefficients of a non-zero Siegel cusp form of arbitrary degree over congruence subgroups. As a corollary of our result for degree two Siegel cusp forms, we get sign changes of its diagonal Fourier coefficients. In the course of our proofs, we prove the non-vanishing of certain type of Fourier-Jacobi coefficients of a Siegel cusp form and all theta components of certain Jacobi cusp forms of arbitrary degree over congruence subgroups, which are also of independent interest.
{"title":"On sign changes of primitive Fourier coefficients of Siegel cusp forms","authors":"K. D. Shankhadhar, P. Tiwari","doi":"10.7169/facm/2101","DOIUrl":"https://doi.org/10.7169/facm/2101","url":null,"abstract":"In this article, we establish quantitative results for sign changes in certain subsequences of primitive Fourier coefficients of a non-zero Siegel cusp form of arbitrary degree over congruence subgroups. As a corollary of our result for degree two Siegel cusp forms, we get sign changes of its diagonal Fourier coefficients. In the course of our proofs, we prove the non-vanishing of certain type of Fourier-Jacobi coefficients of a Siegel cusp form and all theta components of certain Jacobi cusp forms of arbitrary degree over congruence subgroups, which are also of independent interest.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43693186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
are the generalized hyperharmonic numbers (see [4, 10]). Furthermore, H (p,1) n = H (p) n = ∑n j=1 1/n p are the generalized harmonic numbers and H (1,r) n = h (r) n are the classical hyperharmonic numbers. In particularH (1,1) n = Hn are the classical harmonic numbers. Many researchers have been studying Euler sums of harmonic and hyperharmonic numbers (see [4, 6, 7, 9] and references therein), since they play
是普通的超谐波数字(见[4,10])。Furthermore, H (p, 1) n = H (p) n =∑n j = 1 / n p generalized是调和定律数字1和H (r, r) n = H (n)是《古典hyperharmonic数字。特别是特别是许多研究人员自从他们开始演奏以来，一直在研究Euler的和声和超谐波数字(见[4,6,7,9]和therein引用)
{"title":"Euler sums of generalized hyperharmonic numbers","authors":"Rusen Li","doi":"10.7169/facm/1953","DOIUrl":"https://doi.org/10.7169/facm/1953","url":null,"abstract":"are the generalized hyperharmonic numbers (see [4, 10]). Furthermore, H (p,1) n = H (p) n = ∑n j=1 1/n p are the generalized harmonic numbers and H (1,r) n = h (r) n are the classical hyperharmonic numbers. In particularH (1,1) n = Hn are the classical harmonic numbers. Many researchers have been studying Euler sums of harmonic and hyperharmonic numbers (see [4, 6, 7, 9] and references therein), since they play","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47163246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $a_1,cdots,a_6$ be non-zero integers satisfying $(a_i,a_j)=1, 1leq i lt j leq 6$ and $b$ be any integer. For the Diophantine equation $a_1p_1+a_2p_2^3+cdots+a_6p_6^3=b$ we prove that (i) if all $a_1,cdots,a_6$ are positive and $bgg max {|a_j|}^{34+varepsilon}$, then the equation is soluble in primes $p_j$, and (ii) if $a_1,cdots,a_6$ are not all of the same sign, then the equation has prime solutions satisfying $max { p_1,p_2^3,cdots,p_6^3 }ll |b|+max {|a_j|}^{33+varepsilon}$, where the implied constants depend only on $varepsilon$.
设$a_1,cdots,a_6$为非零整数，满足$(a_i,a_j)=1, 1leq i lt j leq 6$, $b$为任意整数。对于Diophantine方程$a_1p_1+a_2p_2^3+cdots+a_6p_6^3=b$，我们证明了(i)如果所有$a_1,cdots,a_6$和$bgg max {|a_j|}^{34+varepsilon}$都是正的，则方程可解为质数$p_j$， (ii)如果$a_1,cdots,a_6$不都是相同的符号，则方程有满足$max { p_1,p_2^3,cdots,p_6^3 }ll |b|+max {|a_j|}^{33+varepsilon}$的质数解，其中隐含常数仅依赖于$varepsilon$。
{"title":"Small prime solutions of a Diophantine equation with one prime and five cubes of primes","authors":"Weiping Li","doi":"10.7169/FACM/1874","DOIUrl":"https://doi.org/10.7169/FACM/1874","url":null,"abstract":"Let $a_1,cdots,a_6$ be non-zero integers satisfying $(a_i,a_j)=1, 1leq i lt j leq 6$ and $b$ be any integer. For the Diophantine equation $a_1p_1+a_2p_2^3+cdots+a_6p_6^3=b$ we prove that (i) if all $a_1,cdots,a_6$ are positive and $bgg max {|a_j|}^{34+varepsilon}$, then the equation is soluble in primes $p_j$, and (ii) if $a_1,cdots,a_6$ are not all of the same sign, then the equation has prime solutions satisfying $max { p_1,p_2^3,cdots,p_6^3 }ll |b|+max {|a_j|}^{33+varepsilon}$, where the implied constants depend only on $varepsilon$.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45672076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the distribution of the least common multiple of positive integers in N ∩ [1 , x ] and related problems. We refine some results of Hilberdink and T´oth (2016). We also give a partial result toward a conjecture of Hilberdink, Luca, and T´oth (2020).
{"title":"On the distribution of the $lcm$ of $k$-tuples and related problems","authors":"Sungjin Kim","doi":"10.7169/facm/2008","DOIUrl":"https://doi.org/10.7169/facm/2008","url":null,"abstract":"We study the distribution of the least common multiple of positive integers in N ∩ [1 , x ] and related problems. We refine some results of Hilberdink and T´oth (2016). We also give a partial result toward a conjecture of Hilberdink, Luca, and T´oth (2020).","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48879660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that there is essentially a unique elliptic curve E defined over a cubic Galois extension K of Q with a K-rational point of order 13 and such that E is not defined over Q.
{"title":"Elliptic curves with a point of order $13$ defined over cyclic cubic fields","authors":"Peter Bruin, M. Derickx, M. Stoll","doi":"10.7169/facm/1945","DOIUrl":"https://doi.org/10.7169/facm/1945","url":null,"abstract":"We show that there is essentially a unique elliptic curve E defined over a cubic Galois extension K of Q with a K-rational point of order 13 and such that E is not defined over Q.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44333860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, Garunkštis, Laurinc̆ikas, Matsumoto, J. & R. Steuding showed an effective universality-type theorem for the Riemann zeta-function by using an effective multidimensional denseness result of Voronin. We will generalize Voronin’s effective result and their theorem to the elements of the Selberg class satisfying some conditions.
{"title":"Effective uniform approximation by $ L$-functionsin the Selberg class","authors":"K. Endo","doi":"10.7169/facm/2026","DOIUrl":"https://doi.org/10.7169/facm/2026","url":null,"abstract":"Recently, Garunkštis, Laurinc̆ikas, Matsumoto, J. & R. Steuding showed an effective universality-type theorem for the Riemann zeta-function by using an effective multidimensional denseness result of Voronin. We will generalize Voronin’s effective result and their theorem to the elements of the Selberg class satisfying some conditions.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49569847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we consider the problem of finding pairs of triangles whose sides are perfect squares of integers, and which have a common perimeter and common area. We find two such pairs of triangles, and prove that there exist infinitely many pairs of triangles with the specified properties. Mathematics Subject Classification 2020: 11D41
{"title":"Pairs of equiperimeter and equiareal triangles whose sides are perfect squares","authors":"A. Choudhry, A. S. Zargar","doi":"10.7169/facm/1985","DOIUrl":"https://doi.org/10.7169/facm/1985","url":null,"abstract":"In this paper we consider the problem of finding pairs of triangles whose sides are perfect squares of integers, and which have a common perimeter and common area. We find two such pairs of triangles, and prove that there exist infinitely many pairs of triangles with the specified properties. Mathematics Subject Classification 2020: 11D41","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44679867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study Fourier and Laplace transforms for Fourier hyperfunctions with values in a complex locally convex Hausdorff space. Since any hyperfunction with values in a wide class of locally convex Hausdorff spaces can be extended to a Fourier hyperfunction, this gives simple notions of asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions, which improves the existing models of Komatsu, Bäumer, Lumer and Neubrander and Langenbruch.
{"title":"Asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions","authors":"K. Kruse","doi":"10.7169/facm/1955","DOIUrl":"https://doi.org/10.7169/facm/1955","url":null,"abstract":"We study Fourier and Laplace transforms for Fourier hyperfunctions with values in a complex locally convex Hausdorff space. Since any hyperfunction with values in a wide class of locally convex Hausdorff spaces can be extended to a Fourier hyperfunction, this gives simple notions of asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions, which improves the existing models of Komatsu, Bäumer, Lumer and Neubrander and Langenbruch.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45746409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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