{"title":"Location of zeros of the polar derivative of a polynomial","authors":"B. Zargar, M. Gulzar, S. A. Malik","doi":"10.7153/jca-2022-19-01","DOIUrl":"https://doi.org/10.7153/jca-2022-19-01","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136499","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":"A note on a family of log-integrals","authors":"Khristo N. Boyadzhiev, R. Frontczak","doi":"10.7153/jca-2022-20-10","DOIUrl":"https://doi.org/10.7153/jca-2022-20-10","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136716","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 Riemann zeta function ζ(s) is one of the most important special functions of Mathematics. While the critical strip 0 < R (s) < 1 is undoubtedly the most important region in the complex plane on account of the unsolved problem regarding location of non-trivial zeros of ζ(s), namely, the Riemann Hypothesis, the right-half plane R (s) > 1 also has its own share of interesting unsolved problems to contribute to. It is quite well known that many number theoretic properties of odd zeta values are nowadays still unsolved mysteries, such as the rationality, transcendence and existence of closed-forms. Only in 1978 did Apéry [2] famously proved that ζ(3) is irrational. This was later reproved in a variety of ways by several authors, in particular Beukers [10] who devised a simple approach involving certain integrals over [0, 1]. In the early 2000s, an important work of Rivoal [21], and Ball and Rivoal [4] determined that infinitely many values of ζ at odd integers are irrational, and the work of Zudilin [27] proved that at least one among ζ(5), ζ(7), ζ(9) and ζ(11) is irrational. A very recent result due to Rivoal and Zudilin [22] states that at least two of the numbers ζ(5), ζ(7), . . . , ζ(69) are irrational. Moreover, for any pair of positive integers a and b, Haynes and Zudilin [17, Theorem 1] have shown that either there are infinitely many m ∈ N for which ζ(am+ b) is irrational, or the sequence {qm} ∞ m=1 of common denominators of the rational elements of the set {ζ(a+ b), . . . , ζ(am+ b)} grows super-exponentially, that is, q 1/m m → ∞ as m → ∞. Despite these advances, to this day no value of ζ(2n+ 1) with n > 2 is known to be irrational. A folklore conjecture states that the numbers π, ζ(3), ζ(5), ζ(7), . . . are algebraically independent over the rationals. This conjecture is predicted by the Grothendieck’s period conjecture for mixed Tate motives. But both conjectures are far out of reach and we do not even know the transcendence of a single odd zeta value. One should mention that Brown [11] has in the past few years outlined a simple geometric approach to understand the structures involved in Beukers’s proof of irrationality of ζ(3) and how this may generalize to other odd zeta values. Ramanujan made many beautiful and elegant discoveries in his short life of 32 years. One of the most remarkable formulas suggested by Ramanujan that has attracted the attention of several mathematicians over the years is the following intriguing identity involving the odd values of the Riemann zeta function [6, 1.2]:
黎曼ζ（s）函数是数学中最重要的特殊函数之一。虽然临界带01也有自己的有趣的未解决的问题。众所周知，奇ζ值的许多数论性质至今仍是未解之谜，如封闭形式的合理性、超越性和存在性。直到1978年，Apéry[2]才著名地证明ζ（3）是非理性的。这后来被几位作者以各种方式谴责，特别是Beukers[10]，他设计了一种涉及[0,1]上某些积分的简单方法。在21世纪初，Rivoal[21]、Ball和Rivoal[4]的一项重要工作确定了奇整数上ζ的无穷多个值是无理的，Zudilin[27]的工作证明了ζ（5）、ζ（7）、ξ（9）和ζ（11）中至少有一个是无理的。Rivoal和Zudilin[22]最近的一个结果表明，至少有两个数字ζ（5），ζ（7），ζ（69）是不合理的。此外，对于任何一对正整数a和b，Haynes和Zudilin[17，定理1]已经证明，要么有无限多个m∈N，其中ζ（am+b）是无理的，要么集合{ζ（a+b），…，ζ（m+b）}的有理元素的公共分母的序列{qm}∞m=1超指数增长，即q1/m→ ∞ as m→ ∞. 尽管取得了这些进展，但到目前为止，还没有发现n＞2的ζ（2n+1）的值是不合理的。一个民间传说推测，数字π，ζ（3），ζ。在代数上独立于有理数。这个猜想是由Grothendieck的泰特混合动机时期猜想预测的。但这两种猜测都遥不可及，我们甚至不知道单个奇ζ值的超越性。值得一提的是，Brown[11]在过去几年中概述了一种简单的几何方法，以理解Beukers证明ζ（3）的非理性所涉及的结构，以及这如何推广到其他奇怪的ζ值。拉马努詹在短短32年的生命中，做出了许多美丽而优雅的发现。Ramanujan提出的最引人注目的公式之一，多年来吸引了几位数学家的注意，是以下涉及黎曼ζ函数奇值的有趣恒等式[6，1.2]：
{"title":"An elementary proof of Ramanujan's identity for odd zeta values","authors":"Sarth Chavan","doi":"10.7153/jca-2022-19-11","DOIUrl":"https://doi.org/10.7153/jca-2022-19-11","url":null,"abstract":"The Riemann zeta function ζ(s) is one of the most important special functions of Mathematics. While the critical strip 0 < R (s) < 1 is undoubtedly the most important region in the complex plane on account of the unsolved problem regarding location of non-trivial zeros of ζ(s), namely, the Riemann Hypothesis, the right-half plane R (s) > 1 also has its own share of interesting unsolved problems to contribute to. It is quite well known that many number theoretic properties of odd zeta values are nowadays still unsolved mysteries, such as the rationality, transcendence and existence of closed-forms. Only in 1978 did Apéry [2] famously proved that ζ(3) is irrational. This was later reproved in a variety of ways by several authors, in particular Beukers [10] who devised a simple approach involving certain integrals over [0, 1]. In the early 2000s, an important work of Rivoal [21], and Ball and Rivoal [4] determined that infinitely many values of ζ at odd integers are irrational, and the work of Zudilin [27] proved that at least one among ζ(5), ζ(7), ζ(9) and ζ(11) is irrational. A very recent result due to Rivoal and Zudilin [22] states that at least two of the numbers ζ(5), ζ(7), . . . , ζ(69) are irrational. Moreover, for any pair of positive integers a and b, Haynes and Zudilin [17, Theorem 1] have shown that either there are infinitely many m ∈ N for which ζ(am+ b) is irrational, or the sequence {qm} ∞ m=1 of common denominators of the rational elements of the set {ζ(a+ b), . . . , ζ(am+ b)} grows super-exponentially, that is, q 1/m m → ∞ as m → ∞. Despite these advances, to this day no value of ζ(2n+ 1) with n > 2 is known to be irrational. A folklore conjecture states that the numbers π, ζ(3), ζ(5), ζ(7), . . . are algebraically independent over the rationals. This conjecture is predicted by the Grothendieck’s period conjecture for mixed Tate motives. But both conjectures are far out of reach and we do not even know the transcendence of a single odd zeta value. One should mention that Brown [11] has in the past few years outlined a simple geometric approach to understand the structures involved in Beukers’s proof of irrationality of ζ(3) and how this may generalize to other odd zeta values. Ramanujan made many beautiful and elegant discoveries in his short life of 32 years. One of the most remarkable formulas suggested by Ramanujan that has attracted the attention of several mathematicians over the years is the following intriguing identity involving the odd values of the Riemann zeta function [6, 1.2]:","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49594737","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 provide new necessary and sufficient conditions for the convergence of positive series developing Bertran–De Morgan and Cauchy type tests given in [M. Martin, Bull. Amer. Math. Soc. 47(1941), 452457] and [L. Bourchtein et al, Int. J. Math. Anal. 6(2012), 1847–1869]. The obtained result enables us to extend the known conditions for recurrence and transience of birth-and-death processes given in [V. M. Abramov, Amer. Math. Monthly 127(2020) 444–448].
本文发展了文献[M]中的Bertran-De Morgan和Cauchy型判别，给出了正级数收敛的新的充分必要条件。马丁,公牛。阿米尔。数学。社会科学，47(1941)，45 - 24 [L]。Bourchtein et al .;j .数学。植物学报，6(2012)，1847-1869。所得结果推广了文献[V]中关于生死过程递归性和暂态性的已知条件。阿布拉莫夫，美国数学。月刊127(2020)444-448]。
{"title":"Necessary and sufficient conditions for the convergence of positive series","authors":"V. Abramov","doi":"10.7153/jca-2022-19-09","DOIUrl":"https://doi.org/10.7153/jca-2022-19-09","url":null,"abstract":"We provide new necessary and sufficient conditions for the convergence of positive series developing Bertran–De Morgan and Cauchy type tests given in [M. Martin, Bull. Amer. Math. Soc. 47(1941), 452457] and [L. Bourchtein et al, Int. J. Math. Anal. 6(2012), 1847–1869]. The obtained result enables us to extend the known conditions for recurrence and transience of birth-and-death processes given in [V. M. Abramov, Amer. Math. Monthly 127(2020) 444–448].","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41922378","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":"Uniform norm estimates of Bernstein-type for lacunary-type complex polyomials","authors":"A. Mir, A. Hussain","doi":"10.7153/jca-2021-18-04","DOIUrl":"https://doi.org/10.7153/jca-2021-18-04","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135899","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":"Integrating the tails of two Maclaurin series","authors":"Russell A. Gordon","doi":"10.7153/jca-2021-18-06","DOIUrl":"https://doi.org/10.7153/jca-2021-18-06","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136090","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 intend to find out some inequalities relating to generalized (α ,β) relative order, generalized (α ,β) relative type and generalized (α ,β) relative weak type of an entire function f with respect to an entire function g when generalized (γ ,β) relative order, generalized (γ ,β) relative type and generalized (γ ,β) relative weak type of f with respect to another entire function h and generalized (γ ,α) relative order, generalized (γ ,α) relative type and generalized (γ ,α) relative weak type of g with respect to h are given, where α , β and γ are continuous non-negative slowly increasing functions defined on (−∞,+∞) . Mathematics subject classification (2020): 30D20, 30D30, 30D35.
{"title":"On some inequalities concerning generalized (α,β) relative order and generalized (α,β) relative type of entire function with respect to an entire function","authors":"T. Biswas, C. Biswas","doi":"10.7153/jca-2021-18-07","DOIUrl":"https://doi.org/10.7153/jca-2021-18-07","url":null,"abstract":"In this paper, we intend to find out some inequalities relating to generalized (α ,β) relative order, generalized (α ,β) relative type and generalized (α ,β) relative weak type of an entire function f with respect to an entire function g when generalized (γ ,β) relative order, generalized (γ ,β) relative type and generalized (γ ,β) relative weak type of f with respect to another entire function h and generalized (γ ,α) relative order, generalized (γ ,α) relative type and generalized (γ ,α) relative weak type of g with respect to h are given, where α , β and γ are continuous non-negative slowly increasing functions defined on (−∞,+∞) . Mathematics subject classification (2020): 30D20, 30D30, 30D35.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136099","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":"On the generalized Hurwitz-Lerch zeta function and generalized Lambert transform","authors":"Viren ra Kumar","doi":"10.7153/jca-2021-17-05","DOIUrl":"https://doi.org/10.7153/jca-2021-17-05","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135779","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}
. An investigation into a family of de fi nite integrals containing log-polylog functions with negative argument will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals and some may be represented as a BBP type formula. In a special case we prove that the corresponding log integral can be represented as a linear combination of the product of zeta functions and the Dirichlet beta function.
{"title":"Alternating Euler sums and BBP-type series","authors":"A. Sofo","doi":"10.7153/jca-2021-18-12","DOIUrl":"https://doi.org/10.7153/jca-2021-18-12","url":null,"abstract":". An investigation into a family of de fi nite integrals containing log-polylog functions with negative argument will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals and some may be represented as a BBP type formula. In a special case we prove that the corresponding log integral can be represented as a linear combination of the product of zeta functions and the Dirichlet beta function.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136385","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":"On weighted β-absolute convergence of double Fourier series","authors":"K. N. Darji, R. Vyas","doi":"10.7153/jca-2021-18-01","DOIUrl":"https://doi.org/10.7153/jca-2021-18-01","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135798","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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