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A note on the Irrationality of $ζ(5)$ and Higher Odd Zeta Values 关于$ζ(5)$及更高奇数Zeta值非理性的说明

arXiv - MATH - General Mathematics Pub Date : 2024-07-08 DOI: arxiv-2407.07121

Shekhar Suman

{"title":"A note on the Irrationality of $ζ(5)$ and Higher Odd Zeta Values","authors":"Shekhar Suman","doi":"arxiv-2407.07121","DOIUrl":"https://doi.org/arxiv-2407.07121","url":null,"abstract":"For $ninmathbb{N}$ we define a double integral begin{equation*}\u0000I_n=frac{1}{24}int_0^1int_0^1 frac{-log^3(xy)}{1+xy} (xy(1-xy))^n \u0000dxdyend{equation*} We denote $d_n=text{lcm}(1,2,...,n)$ and prove that for\u0000all $ninmathbb{N}$, begin{equation*} d_n I_n= 15 (-1)^n 2^{n-4} d_n\u0000zeta(5)+(-1)^{n+1} d_n sum_{r=0}^{n} binom{n}{r}left(sum_{k=1}^{n+r}\u0000frac{(-1)^{k-1}}{k^5}right) end{equation*} Now if $zeta(5)$ is rational,\u0000then $zeta(5)=a/b$, $(a,b)=1$ and $a,binmathbb{N}$. Then we take $ngeq b$\u0000so that $b|d_n$. We show that for all $ngeq 1$, $0<d_n I_n<1$. We denote\u0000begin{equation*} S_n=d_n sum_{r=0}^{n} binom{n}{r}left(sum_{k=1}^{n+r}\u0000frac{(-1)^{k-1}}{k^5}right) end{equation*} We prove for all $ngeq b$,\u0000begin{equation*} d_n I_n+(-1)^{n}\u0000left{S_nright}inmathbb{Z}end{equation*} where ${x}=x-[x]$ denotes the\u0000fractional part of $x$ and $[x]$ denotes the greatest integer less than or\u0000equal to $x$. Later we show that the only integer value admissible is\u0000begin{equation*} d_n I_n+(-1)^{n} left{S_nright}=0 text{or} d_n\u0000I_n+(-1)^{n} left{S_nright}=1end{equation*} Finally we prove that these\u0000above values are not possible. This gives a contradiction to our assumption\u0000that $zeta(5)$ is rational. Similarly for all $mgeq 3$ and $ngeq 1$, by\u0000defining begin{equation*} I_{n,m}=frac{1}{Gamma(2m+1)}int_0^1int_0^1\u0000frac{-log^{2m-1}(xy)}{1+xy} (xy(1-xy))^n dxdyend{equation*} where\u0000$Gamma(s)$ denotes the Gamma function, we give an extension of the above proof\u0000to prove the irrationality of higher odd zeta values $zeta(2m+1)$ for all\u0000$mgeq 3$","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584702","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}

引用次数: 0

Describing chaotic systems 描述混沌系统

arXiv - MATH - General Mathematics Pub Date : 2024-07-07 DOI: arxiv-2407.07919

Brandon Le

引用次数: 0

An approach to Borwein integrals from the point of view of residue theory 从残差理论的角度研究博文积分的方法

arXiv - MATH - General Mathematics Pub Date : 2024-07-07 DOI: arxiv-2407.15856

Daniel Cao Labora, Gonzalo Cao Labora

引用次数: 0

Generalization of Cantor Pairing Polynomials (Bijective Mapping Among Natural Numbers) from N02 to N0 to Z2 to N0 and N03 to N 从 N02 到 N0 到 Z2 到 N0 和 N03 到 N 的康托配对多项式（自然数之间的双射映射）的广义化

arXiv - MATH - General Mathematics Pub Date : 2024-07-06 DOI: arxiv-2407.05073

Sandor Kristyan

{"title":"Generalization of Cantor Pairing Polynomials (Bijective Mapping Among Natural Numbers) from N02 to N0 to Z2 to N0 and N03 to N","authors":"Sandor Kristyan","doi":"arxiv-2407.05073","DOIUrl":"https://doi.org/arxiv-2407.05073","url":null,"abstract":"The Cantor pairing polynomials are extended to larger 2D sub-domains and more\u0000complex mapping, of which the most important property is the bijectivity. If\u0000corners are involved inside (but not the borders of) domain, more than one\u0000connected polynomials are necessary. More complex patterns need more complex\u0000subsequent application of math series to obtain the mapping polynomials which\u0000are more and more inconvenient, although elementary. A tricky polynomial fit is\u0000introduced (six coefficients are involved like in the original Cantor\u0000polynomials with rigorous but simple restrictions on points chosen) to buy out\u0000the regular treatment of math series to find the pairing polynomials instantly.\u0000The original bijective Cantor polynomial C1(x,y)= (x2+2xy+y2+3x+y)/2: N02 to N0\u0000(=positive integers) which is 2-fold and runs in zig - zag way along lines\u0000x+y=N is extended e.g. to the bijective P(x,y)=\u00002x2+4sgn(x)sgn(y)xy+2y2-2H(x)sgn(y)x-y+1: Z2 to N0 (with sign and Heaviside\u0000functions, Z is integers) running in spiral way along concentric rhombuses, or\u0000to the bijective P3D(x,y,z)= [x3+y3+z3 +3(xz2+yz2 +zx2+2xyz +zy2+yx2+xy2)\u0000+3(2x2+2y2 +z2+2xz +2yz+4xy) +5x+11y+2z]/6: N03 to N0 which is 6-fold and runs\u0000along plains x+y+z=N. Storage device for triangle matrices is also commented as\u0000cutting the original Cantor domain to half along with related Diophantine\u0000equations.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570922","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}

引用次数: 0

On the Simple Divisibility Restrictions by Polynomial Equation a n+bn=cn Itself in Fermat Last Theorem for Integer/Complex/Quaternion Triples 论整数/复数/四元数三元组费马最后定理中的多项式方程 a n+bn=cn 本身的简单可分性限制

arXiv - MATH - General Mathematics Pub Date : 2024-07-06 DOI: arxiv-2407.05068

Sandor Kristyan

{"title":"On the Simple Divisibility Restrictions by Polynomial Equation a n+bn=cn Itself in Fermat Last Theorem for Integer/Complex/Quaternion Triples","authors":"Sandor Kristyan","doi":"arxiv-2407.05068","DOIUrl":"https://doi.org/arxiv-2407.05068","url":null,"abstract":"The divisibility restrictions in the famous equation a n+bn=cn in Fermat Last\u0000Theorem (FLT, 1637) is analyzed how it selects out many triples to be Fermat\u0000triple (i.e. solutions) if n greater than 2, decreasing the cardinality of\u0000Fermat triples. In our analysis, the restriction on positive integer (PI)\u0000solutions ((a,b,c,n) up to the point when there is no more) is not along with\u0000restriction on power n in PI as decreasing sets {PI } containing {odd}\u0000containing {primes} containing {regular primes}, etc. as in the literature, but\u0000with respect to exclusion of more and more c in PI as increasing sets {primes\u0000p} in {p k} in {PI}. The divisibility and co-prime property in Fermat equation\u0000is analyzed in relation to exclusion of solutions, and the effect of\u0000simultaneous values of gcd(a,b,c), gcd(a+b,cn), gcd(c-a,bn) and gcd(c-b,an) on\u0000the decrease of cardinality of solutions is exhibited. Again, our derivation\u0000focuses mainly on the variable c rather than on variable n, oppositely to the\u0000literature in which the FLT is historically separated via the values of power\u0000n. Among the most famous are the known, about 2500 years old, existing\u0000Pythagorean triples (a,b,c,n=2) and the first milestones as the proved cases\u0000(of non-existence as n=3 by Gauss and later by Euler (1753) and n=4 by Fermat)\u0000less than 400 years ago. As it is known, Wiles has proved the FLT in 1995 in an\u0000abstract roundabout way. The n<0, n:=1/m, as well as complex and quaternion\u0000(a,b,c) cases focusing on Pythagoreans are commented. Odd powers FLT over\u0000quaternions breaks.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570921","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}

引用次数: 0

Graph Linear Canonical Transform: Definition, Vertex-Frequency Analysis and Filter Design 图线性典型变换：定义、顶点频率分析和滤波器设计

arXiv - MATH - General Mathematics Pub Date : 2024-07-02 DOI: arxiv-2407.12046

Jian Yi Chen, Bing Zhao Li

引用次数: 0

About zero counting of Riemann Z function 关于黎曼 Z 函数的零点计数

arXiv - MATH - General Mathematics Pub Date : 2024-07-02 DOI: arxiv-2407.07910

Giovanni Lodone

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Baker's dozen digits of two sums involving reciprocal products of an integer and its greatest prime factor 涉及整数及其最大质因数互积的两个和的贝克十几位数

arXiv - MATH - General Mathematics Pub Date : 2024-07-02 DOI: arxiv-2407.12047

Tengiz O. Gogoberidze

引用次数: 0

Variance of the distance to the boundary of convex domains in $mathbb{R}^{2}$ and $mathbb{R}^{3}$ $mathbb{R}^{2}$和$mathbb{R}^{3}$中凸域边界距离的方差

arXiv - MATH - General Mathematics Pub Date : 2024-07-01 DOI: arxiv-2407.12041

Alastair N. Fletcher, Alexander G. Fletcher

引用次数: 0

On Bounds and Diophantine Properties of Elliptic Curves 论椭圆曲线的边界和 Diophantine 特性

arXiv - MATH - General Mathematics Pub Date : 2024-06-30 DOI: arxiv-2407.09558

Navvye Anand

引用次数: 0