By the theory of elliptic curves, we show that there are infinitely many integral right triangle-perpendicular quadrilateral, integral isosceles triangle-perpendicular quadrilateral, and Heron triangle-perpendicular quadrilateral pairs with a common area and a common perimeter. Moreover, for the elliptic curve associated to integral isosceles triangle and integral perpendicular quadrilateral pairs, we present several subfamilies of rank $geq 4$, and show the existence of infinitely many elliptic curves of rank $geq 5$, parameterized by the points of an elliptic curve of positive rank.
{"title":"Integral triangles and perpendicular quadrilateral pairs with a common area and a common perimeter","authors":"A. S. Zargar, Yong Zhang","doi":"10.7169/facm/1842","DOIUrl":"https://doi.org/10.7169/facm/1842","url":null,"abstract":"By the theory of elliptic curves, we show that there are infinitely many integral right triangle-perpendicular quadrilateral, integral isosceles triangle-perpendicular quadrilateral, and Heron triangle-perpendicular quadrilateral pairs with a common area and a common perimeter. Moreover, for the elliptic curve associated to integral isosceles triangle and integral perpendicular quadrilateral pairs, we present several subfamilies of rank $geq 4$, and show the existence of infinitely many elliptic curves of rank $geq 5$, parameterized by the points of an elliptic curve of positive rank.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46205215","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 discuss evaluating fractional Stieltjes constants γα(a), arising naturally from the Laurent series expansions of the fractional derivatives of the Hurwitz zeta functions ζ(α)(s, a). We give an upper bound for the absolute value of Cα(a) = γα(a) − log(a)/a and an asymptotic formula C̃α(a) for Cα(a) that yields a good approximation even for most small values of α. We bound |C̃α(a)| and based on this conjecture a tighter bound for |Cα(a)|
{"title":"Approximating and bounding fractional Stieltjes constants","authors":"Ricky E. Farr, S. Pauli, F. Saidak","doi":"10.7169/facm/1868","DOIUrl":"https://doi.org/10.7169/facm/1868","url":null,"abstract":"We discuss evaluating fractional Stieltjes constants γα(a), arising naturally from the Laurent series expansions of the fractional derivatives of the Hurwitz zeta functions ζ(α)(s, a). We give an upper bound for the absolute value of Cα(a) = γα(a) − log(a)/a and an asymptotic formula C̃α(a) for Cα(a) that yields a good approximation even for most small values of α. We bound |C̃α(a)| and based on this conjecture a tighter bound for |Cα(a)|","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47499838","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}
A Delone set in $mathbb{R}^n$ is a set such that (a) the distance between any two of its points is uniformly bounded below by a strictly positive constant and such that (b) the distance from any point to the remaining points in the set is uniformly bounded above. Delone sets are thus sets of points enjoying nice spacing properties, and appear therefore naturally in mathematical models for quasicrystals. Define a spiral set in $mathbb{R}^n$ as a set of points of the form $left{sqrt[n]{k}cdotboldsymbol{u}_kright}_{kge 1}$, where $left(boldsymbol{u}_kright)_{kge 1}$ is a sequence in the unit sphere $mathbb{S}^{n-1}$. In the planar case $n=2$, spiral sets serve as natural theoretical models in phyllotaxis (the study of configurations of leaves on a plant stem), and an important example in this class includes the sunflower spiral. Recent works by Akiyama, Marklof and Yudin provide a reasonable complete characterisation of planar spiral sets which are also Delone. A related problem that has emerged in several places in the literature over the past fews years is to determine whether this theory can be extended to higher dimensions, and in particular to show the existence of spiral Delone sets in any dimension. This paper addresses this question by characterising the Delone property of a spiral set in terms of packing and covering conditions satisfied by the spherical sequence $left(boldsymbol{u}_kright)_{kge 1}$. This allows for the construction of explicit examples of spiral Delone sets in $mathbb{R}^n$ for all $nge 2$, which boils down to finding a sequence of points in $mathbb{S}^{n-1}$ enjoying some optimal distribution properties.
{"title":"Higher dimensional spiral Delone sets","authors":"F. Adiceam, Ioannis Tsokanos","doi":"10.7169/facm/1958","DOIUrl":"https://doi.org/10.7169/facm/1958","url":null,"abstract":"A Delone set in $mathbb{R}^n$ is a set such that (a) the distance between any two of its points is uniformly bounded below by a strictly positive constant and such that (b) the distance from any point to the remaining points in the set is uniformly bounded above. Delone sets are thus sets of points enjoying nice spacing properties, and appear therefore naturally in mathematical models for quasicrystals. \u0000Define a spiral set in $mathbb{R}^n$ as a set of points of the form $left{sqrt[n]{k}cdotboldsymbol{u}_kright}_{kge 1}$, where $left(boldsymbol{u}_kright)_{kge 1}$ is a sequence in the unit sphere $mathbb{S}^{n-1}$. In the planar case $n=2$, spiral sets serve as natural theoretical models in phyllotaxis (the study of configurations of leaves on a plant stem), and an important example in this class includes the sunflower spiral. \u0000Recent works by Akiyama, Marklof and Yudin provide a reasonable complete characterisation of planar spiral sets which are also Delone. A related problem that has emerged in several places in the literature over the past fews years is to determine whether this theory can be extended to higher dimensions, and in particular to show the existence of spiral Delone sets in any dimension. \u0000This paper addresses this question by characterising the Delone property of a spiral set in terms of packing and covering conditions satisfied by the spherical sequence $left(boldsymbol{u}_kright)_{kge 1}$. This allows for the construction of explicit examples of spiral Delone sets in $mathbb{R}^n$ for all $nge 2$, which boils down to finding a sequence of points in $mathbb{S}^{n-1}$ enjoying some optimal distribution properties.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43876060","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}
Under special conditions, we prove that the set of preperiodic points for semigroups of self-morphisms of affine spaces falling on cyclotomic closures is not dense. generalising results of Ostafe and Young (2020). We also extend previous results about boundness of house and height on certain preperiodicity sets of higher dimension in semigroup dynamics.
{"title":"Cyclotomic preperiodic points for morphismsin affine spaces and preperiodic points with bounded house and height","authors":"J. Mello","doi":"10.7169/facm/2022","DOIUrl":"https://doi.org/10.7169/facm/2022","url":null,"abstract":"Under special conditions, we prove that the set of preperiodic points for semigroups of self-morphisms of affine spaces falling on cyclotomic closures is not dense. generalising results of Ostafe and Young (2020). We also extend previous results about boundness of house and height on certain preperiodicity sets of higher dimension in semigroup dynamics.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49116930","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 $mathfrak{q}>2$ be a prime number, $chi$ a primitive Dirichlet character modulo $mathfrak{q}$ and $f$ a primitive holomorphic cusp form or a Hecke-Maass cusp form of level $mathfrak{q}$ and trivial nebentypus. We prove the subconvex bound $$ L(1/2,fotimes chi)ll mathfrak{q}^{1/2-1/12+varepsilon}, $$ where the implicit constant depends only on the archimedean parameter of $f$ and $varepsilon$. The main input is a modifying trivial delta method developed in [1].
{"title":"A subconvex bound for twisted $L$-functions","authors":"Qingfeng Sun, Hui Wang","doi":"10.7169/facm/1940","DOIUrl":"https://doi.org/10.7169/facm/1940","url":null,"abstract":"Let $mathfrak{q}>2$ be a prime number, $chi$ a primitive Dirichlet character modulo $mathfrak{q}$ and $f$ a primitive holomorphic cusp form or a Hecke-Maass cusp form of level $mathfrak{q}$ and trivial nebentypus. We prove the subconvex bound $$ L(1/2,fotimes chi)ll mathfrak{q}^{1/2-1/12+varepsilon}, $$ where the implicit constant depends only on the archimedean parameter of $f$ and $varepsilon$. The main input is a modifying trivial delta method developed in [1].","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141204472","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 first moments of central values of Hecke $L$-functions associated with quadratic, cubic and quartic symbols to prime moduli. This also enables us to obtain results on first moments of central values of certain families of cubic and quartic Dirichlet $L$-functions of prime moduli.
{"title":"First moments of some Hecke $L$-functions of prime moduli","authors":"Peng Gao, Liangyi Zhao","doi":"10.7169/facm/1936","DOIUrl":"https://doi.org/10.7169/facm/1936","url":null,"abstract":"We study the first moments of central values of Hecke $L$-functions associated with quadratic, cubic and quartic symbols to prime moduli. This also enables us to obtain results on first moments of central values of certain families of cubic and quartic Dirichlet $L$-functions of prime moduli.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46079408","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}
The second author studied arithmetic properties of a class of sequences that generalize the sequence of derangements. The aim of the following paper is to disprove two conjectures stated in cite{miska}. The first conjecture regards the set of prime divisors of their terms. The latter one is devoted to the order of magnitude of considered sequences.
{"title":"On two conjectures regarding generalized sequence of derangements","authors":"Eryk Lipka, Piotr Miska","doi":"10.7169/facm/1989","DOIUrl":"https://doi.org/10.7169/facm/1989","url":null,"abstract":"The second author studied arithmetic properties of a class of sequences that generalize the sequence of derangements. The aim of the following paper is to disprove two conjectures stated in cite{miska}. The first conjecture regards the set of prime divisors of their terms. The latter one is devoted to the order of magnitude of considered sequences.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44218504","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 give a correspondence between automorphic pairs of distributions on $mathbb{R}$ and Dirichlet series satisfying functional equations and some additional analytic conditions. Moreover, we show that the notion of automorphic pairs of distributions on $mathbb{R}$ can be regarded as a generalization of automorphic distributions on smooth principal series representations of the universal covering group of $SL(2,mathbb{R})$. As an application, we prove Weil type converse theorems for automorphic distributions and Maass forms of real weights.
{"title":"Automorphic pairs of distributions on $mathbb{R}$ and Maass forms of real weight","authors":"T. Miyazaki","doi":"10.7169/facm/1990","DOIUrl":"https://doi.org/10.7169/facm/1990","url":null,"abstract":"We give a correspondence between automorphic pairs of distributions on $mathbb{R}$ and Dirichlet series satisfying functional equations and some additional analytic conditions. Moreover, we show that the notion of automorphic pairs of distributions on $mathbb{R}$ can be regarded as a generalization of automorphic distributions on smooth principal series representations of the universal covering group of $SL(2,mathbb{R})$. As an application, we prove Weil type converse theorems for automorphic distributions and Maass forms of real weights.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45760085","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}
A BSTRACT . We study multiplicative functions f satisfying | f ( n ) | ≤ 1 for all n , the associated Dirichlet series F ( s ) : = P ∞ n = 1 f ( n ) n − s , and the summatory function S f ( x ) : = P n ≤ x f ( n ) . Up to a possible trivial contribution from the numbers f (2 k ) , F ( s ) may have at most one zero or one pole on the one-line, in a sense made precise by Halász. We estimate log F ( s ) away from any such point and show that if F ( s ) has a zero on the one-line in the sense of Halász, then | S f ( x ) | ≤ ( x /log x )exp ¡ c p loglog x ¢ for all c > 0 when x is large enough. This bound is best possible.
摘要。我们研究了所有n满足|f（n）|≤1的乘性函数f，相关的Dirichlet级数f（s）：=P∞n=1f（n，n−s，以及求和函数Sf（x）：=Pn≤xf（n）。在Halász精确指出的意义上，在数字f（2k）的可能微不足道的贡献下，f（s）在一条线上最多可能有一个零或一个极点。我们估计了远离任何这样的点的log F（s），并证明如果F（s）在Halász意义上的一条线上有一个零，那么当x足够大时，|SF（x）|≤（x/logx）expéc p loglog x¢对于所有c>0。这个界限是最好的可能。
{"title":"A footnote to a theorem of Halász","authors":"'Eric Saias, K. Seip","doi":"10.7169/facm/1847","DOIUrl":"https://doi.org/10.7169/facm/1847","url":null,"abstract":"A BSTRACT . We study multiplicative functions f satisfying | f ( n ) | ≤ 1 for all n , the associated Dirichlet series F ( s ) : = P ∞ n = 1 f ( n ) n − s , and the summatory function S f ( x ) : = P n ≤ x f ( n ) . Up to a possible trivial contribution from the numbers f (2 k ) , F ( s ) may have at most one zero or one pole on the one-line, in a sense made precise by Halász. We estimate log F ( s ) away from any such point and show that if F ( s ) has a zero on the one-line in the sense of Halász, then | S f ( x ) | ≤ ( x /log x )exp ¡ c p loglog x ¢ for all c > 0 when x is large enough. This bound is best possible.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46533920","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}
For an extension $K/mathbb{F}_q(T)$ of the rational function field over a finite field, we introduce the notion of virtually $K$-rational Drinfeld modules as a function field analogue of $mathbb{Q}$-curves. Our goal in this article is to prove that all virtually $K$-rational Drinfeld modules of rank two with no complex multiplication are parametrized up to isogeny by $K$-rational points of a quotient curve of the Drinfeld modular curve $Y_0(mathfrak{n})$ with some square-free level $mathfrak{n}$. This is an analogue of Elkies' well-known result on $mathbb{Q}$-curves.
{"title":"Parametrization of virtually $K$-rational Drinfeld modules of rank two","authors":"Y. Okumura","doi":"10.7169/facm/1905","DOIUrl":"https://doi.org/10.7169/facm/1905","url":null,"abstract":"For an extension $K/mathbb{F}_q(T)$ of the rational function field over a finite field, we introduce the notion of virtually $K$-rational Drinfeld modules as a function field analogue of $mathbb{Q}$-curves. Our goal in this article is to prove that all virtually $K$-rational Drinfeld modules of rank two with no complex multiplication are parametrized up to isogeny by $K$-rational points of a quotient curve of the Drinfeld modular curve $Y_0(mathfrak{n})$ with some square-free level $mathfrak{n}$. This is an analogue of Elkies' well-known result on $mathbb{Q}$-curves.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49013875","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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