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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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}
{"title":"The Okada space and vanishing of ${L(1,{f})}$","authors":"M. Murty, Siddhi Pathak","doi":"10.7169/facm/1952","DOIUrl":"https://doi.org/10.7169/facm/1952","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42035749","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 the Minkowski question mark function ?(x) we consider derivative of the function fn(x) = ?(?(...? } {{ } n times (x))). Apart from obvious cases (rational numbers for example) it is non-trivial to find explicit examples of numbers x for which f ′ n (x) = 0. In this paper we present a set of irrational numbers, such that for every element x0 of this set and for any n ∈ Z+ one has f ′ n (x0) = 0.
{"title":"On the derivative of iterations of the Minkowski question mark function at special points","authors":"N. Shulga","doi":"10.7169/facm/1966","DOIUrl":"https://doi.org/10.7169/facm/1966","url":null,"abstract":"For the Minkowski question mark function ?(x) we consider derivative of the function fn(x) = ?(?(...? } {{ } n times (x))). Apart from obvious cases (rational numbers for example) it is non-trivial to find explicit examples of numbers x for which f ′ n (x) = 0. In this paper we present a set of irrational numbers, such that for every element x0 of this set and for any n ∈ Z+ one has f ′ n (x0) = 0.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47218659","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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