Integral and Series Representations of Riemann's Zeta Function And
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Limiting Values and Functional and Difference Equations †
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 February 2020 doi:10.20944/preprints202002.0245.v1 Peer-reviewed version available at Mathematics 2020, 8, 407; doi:10.3390/math8030407 Article Limiting values and functional and difference equations † N. -L. Wang 1, P. Agarwal 2,* and S. Kanemitsu 3 1 College of Applied Mathematics and Computer Science, ShangLuo University, Shangluo, 726000 Shaanxi, P.R. China; E-mail:[email protected] 2 Anand International College of Engineering, Near Kanota, Agra Road, Jaipur-303012, Rajasthan, India 3 Faculty of Engrg Kyushu Inst. Tech. 1-1Sensuicho Tobata Kitakyushu 804-8555, Japan; [email protected] * Correspondence: [email protected] † Dedicated to Professor Dr. Yumiko Hironaka with great respect and friendship Version February 17, 2020 submitted to Journal Not Specified Abstract: Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function. Keywords: limit values; modular relation; Lerch zeta-function; Hurwitz zeta-function; Laurent coefficients MSC: 11F03; 01A55; 40A30; 42A16 1. Introduction There have appeared enormous amount of papers on the Laurent coefficients of a large class of zeta-, L- and special functions. -
Of the Riemann Hypothesis
A Simple Counterexample to Havil's \Reformulation" of the Riemann Hypothesis Jonathan Sondow 209 West 97th Street New York, NY 10025 [email protected] The Riemann Hypothesis (RH) is \the greatest mystery in mathematics" [3]. It is a conjecture about the Riemann zeta function. The zeta function allows us to pass from knowledge of the integers to knowledge of the primes. In his book Gamma: Exploring Euler's Constant [4, p. 207], Julian Havil claims that the following is \a tantalizingly simple reformulation of the Rie- mann Hypothesis." Havil's Conjecture. If 1 1 X (−1)n X (−1)n cos(b ln n) = 0 and sin(b ln n) = 0 na na n=1 n=1 for some pair of real numbers a and b, then a = 1=2. In this note, we first state the RH and explain its connection with Havil's Conjecture. Then we show that the pair of real numbers a = 1 and b = 2π=ln 2 is a counterexample to Havil's Conjecture, but not to the RH. Finally, we prove that Havil's Conjecture becomes a true reformulation of the RH if his conclusion \then a = 1=2" is weakened to \then a = 1=2 or a = 1." The Riemann Hypothesis In 1859 Riemann published a short paper On the number of primes less than a given quantity [6], his only one on number theory. Writing s for a complex variable, he assumes initially that its real 1 part <(s) is greater than 1, and he begins with Euler's product-sum formula 1 Y 1 X 1 = (<(s) > 1): 1 ns p 1 − n=1 ps Here the product is over all primes p. -
The Riemann and Hurwitz Zeta Functions, Apery's Constant and New
The Riemann and Hurwitz zeta functions, Apery’s constant and new rational series representations involving ζ(2k) Cezar Lupu1 1Department of Mathematics University of Pittsburgh Pittsburgh, PA, USA Algebra, Combinatorics and Geometry Graduate Student Research Seminar, February 2, 2017, Pittsburgh, PA A quick overview of the Riemann zeta function. The Riemann zeta function is defined by 1 X 1 ζ(s) = ; Re s > 1: ns n=1 Originally, Riemann zeta function was defined for real arguments. Also, Euler found another formula which relates the Riemann zeta function with prime numbrs, namely Y 1 ζ(s) = ; 1 p 1 − ps where p runs through all primes p = 2; 3; 5;:::. A quick overview of the Riemann zeta function. Moreover, Riemann proved that the following ζ(s) satisfies the following integral representation formula: 1 Z 1 us−1 ζ(s) = u du; Re s > 1; Γ(s) 0 e − 1 Z 1 where Γ(s) = ts−1e−t dt, Re s > 0 is the Euler gamma 0 function. Also, another important fact is that one can extend ζ(s) from Re s > 1 to Re s > 0. By an easy computation one has 1 X 1 (1 − 21−s )ζ(s) = (−1)n−1 ; ns n=1 and therefore we have A quick overview of the Riemann function. 1 1 X 1 ζ(s) = (−1)n−1 ; Re s > 0; s 6= 1: 1 − 21−s ns n=1 It is well-known that ζ is analytic and it has an analytic continuation at s = 1. At s = 1 it has a simple pole with residue 1. -
Special Values for the Riemann Zeta Function
Journal of Applied Mathematics and Physics, 2021, 9, 1108-1120 https://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352 Special Values for the Riemann Zeta Function John H. Heinbockel Old Dominion University, Norfolk, Virginia, USA How to cite this paper: Heinbockel, J.H. Abstract (2021) Special Values for the Riemann Zeta Function. Journal of Applied Mathematics The purpose for this research was to investigate the Riemann zeta function at and Physics, 9, 1108-1120. odd integer values, because there was no simple representation for these re- https://doi.org/10.4236/jamp.2021.95077 sults. The research resulted in the closed form expression Received: April 9, 2021 n 21n+ (2n) (−−4) π E2n 2ψ ( 34) Accepted: May 28, 2021 ζ (2nn+= 1) ,= 1,2,3, 21nn++ 21− Published: May 31, 2021 2( 2 12)( n) ! for representing the zeta function at the odd integer values 21n + for n a Copyright © 2021 by author(s) and Scientific Research Publishing Inc. positive integer. The above representation shows the zeta function at odd This work is licensed under the Creative positive integers can be represented in terms of the Euler numbers E2n and Commons Attribution International ψ (2n) 34 License (CC BY 4.0). the polygamma functions ( ) . This is a new result for this study area. http://creativecommons.org/licenses/by/4.0/ For completeness, this paper presents a review of selected properties of the Open Access Riemann zeta function together with how these properties are derived. This paper will summarize how to evaluate zeta (n) for all integers n different from 1. -
On the Values of a Class of Dirichlet Series at Rational Arguments
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 4, April 2010, Pages 1223–1230 S 0002-9939(09)10171-5 Article electronically published on December 4, 2009 ON THE VALUES OF A CLASS OF DIRICHLET SERIES AT RATIONAL ARGUMENTS K. CHAKRABORTY, S. KANEMITSU, AND H.-L. LI (Communicated by Ken Ono) Dedicated to Professor Eiichi Bannai on his sixtieth birthday, with great respect and friendship Abstract. In this paper we shall prove that the combination of the gen- eral distribution property and the functional equation for the Lipschitz-Lerch transcendent capture the whole spectrum of deeper results on the relations be- tween the values at rational arguments of functions of a class of zeta-functions. By Theorem 1 and its corollaries, we can cover all the previous results in a rather simple and lucid way. By considering the limiting cases, we can also deduce new striking identities for Milnor’s gamma functions, among which is the Gauss second formula for the digamma function. 1. Introduction and the finite value case Srivastava [S2] (cf. also Srivastava and Choi [SC], pp. 336–344) gives two simpler proofs of interesting results of Cvijovi´c and Klinowski [CK1] on the finite expression for the values of a Bernoulli polynomial at rational arguments in terms of the values of the Hurwitz zeta-function at integer arguments with rational values for the perturbation. The first proof uses the Fourier series for the periodic Bernoulli polynomial and the formula for the decomposition into residue classes. The second proof uses the functional equation for the partial zeta-function q p 2Γ (s) πs 2πap a (1.1) ζ 1 − s, = cos − ζ s, q (2πq)s 2 q q a=1 ([SC, (8), p. -
An Elementary and Real Approach to Values of the Riemann Zeta Function
An Elementary and Real Approach to Values of the Riemann Zeta Function∗ Armen Bagdasaryan † An elementary approach for computing the values at nega- tive integers of the Riemann zeta function is presented. The approach is based on a new method for ordering the integers and a new method for summation of divergent series. We show that the values of the Riemann zeta function can be computed, without using the theory of analytic continuation and functions of complex variable. MSC numbers: 11M06, 11B68, 40C15 PACS numbers: 02.10.De, 02.30.Lt 1 Introduction The Riemann zeta function ζ(s) is one of the most important objects in the study of number theory (if not in the whole modern mathematics). It is a classical and well-known result that ζ(s), originally defined on the half plane ℜ(s) > 1, can be analytically continued to a meromorphic function on the entire complex plane with the only pole at s = 1, which is a simple pole with residue 1. arXiv:0812.1878v4 [math.NT] 9 Apr 2010 One of the main reasons of interest to ζ(s) is that the special values of ζ(s) at integers have been proved or conjectured to have significant arithmetic meanings. For instance, Zagier’s [40, 41] conjecture concerning the relation between ζ(n) and the n-logarithms for n ≥ 2 and Lichtenbaum’s [26] conjecture connecting ζ(n) with motivic cohomology. ∗extended version of the contribution to the proceedings of the XXVII International Colloquium on Group Theoretical Methods in Physics, Yerevan, Armenia, August 13-19, 2008 †Institution of the Russian Academy of Sciences, V.A. -
Identities for Catalan's Constant Arising from Integrals Depending
Identities for Catalan’s constant arising from integrals depending on a parameter Federica Ferretti∗Alessandro Gambini†and Daniele Ritelli‡ Abstract In this paper we provide some relationships between Catalan’s constant and the 3F2 and 4F3 hypergeomet- ric functions, deriving them from some parametric integrals. In particular, using the complete elliptic integral of the first kind, we found an alternative proof of a result of Ramanujan for 3F2, a second identity related to 4F3 and using the complete elliptic integral of the second kind we obtain an identity by Adamchik. KEYWORD: Catalan constant, elliptic integral, hypergeometric functions. AMS SUBJECT CLASSIFICATION 33C75, 33C20 1 Introduction Catalan’s constant G was defined by Eug`ene Charles Catalan, who introduced this constant in [13, eq. (4) p. 23] as the alternating series ∞ ( 1)n G = ∑ − 2 . n=0 (2n + 1) It is well-known, [14, eq. (1) p. 1], that we may also express this constant with the following definite integral 1 arctan x dx. x Z0 This integral has stimulated the interest of Ramanujan, [23]. G was also identified by James W.L. Glaisher in 1877, see [17]. Its value is approximately G ∼= 0.915965594177 . and actually it is not known if it is a rational number. The constant G is somewhat ubiquitous since it appears in many occurrences connected to definite integrals or series summations; we give a (surely) non exhaustive list of papers that provide several interesting occurrences of G, including [1, 2, 7, 10, 15, 19, 22] and [26]. There are also interesting connections between Catalan’s constant and the Clausen’s integrals, that is arXiv:2005.04672v1 [math.CA] 10 May 2020 θ t Cl2(θ) = log 2 sin dt, − 0 2 Z the Hurwitz zeta function, Euler-Mascheroni constant and many other special functions, as one can see for instance in [10], [15], [26]. -
Special Functions Related to Dedekind Type DC-Sums and Their
SPECIAL FUNCTIONS RELATED TO DEDEKIND TYPE DC-SUMS AND THEIR APPLICATIONS Yilmaz SIMSEK Akdeniz University, Faculty of Arts and Science, Department of Mathematics 07058 Antalya, Turkey [email protected] Abstract In this paper we construct trigonometric functions of the sum Tp(h, k), which is called Dedekind type DC-(Dahee and Changhee) sums. We establish analytic properties of this sum. We find trigonometric represen- tations of this sum. We prove reciprocity theorem of this sums. Furthermore, we obtain relations between the Clausen functions, Polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), Hardy-Berndt sums and the sum Tp(h, k). We also give some applications related to these sums and functions. 2000 Mathematics Subject Classification. 11F20, 11B68, 11M35, 11M41, 11S80, 33C10, 33E20. Key Words and Phrases. Dedekind sums, Hardy-Berndt sums, Bernoulli Functions, Euler Functions, Euler numbers and polynomials, Riemann zeta function, Hurwitz zeta function, Lerch zeta function, Dirichlet series for the polylogarithm function, Dirichlet’s eta function, Legendre chi function, trigonometric functions and generalized Lambert series. arXiv:0902.0380v1 [math.CV] 2 Feb 2009 1 Introduction, Definitions and Notations In this section, we give some definitions, notations and results related to the Dedekind sums. Firstly we start with the definition of the classical Dedekind sums. Let x [x] 1 , if x / Z ((x)) = − G − 2 ∈ 0, if x Z, ∈ 1 [x]G being the largest integer x. Let h and k be coprime integers with k > 0, the classical Dedekind sum≤ s(h, k) is defined as follows k−1 a ha s (h, k)= . k k a=1 X The reciprocity law of the classical Dedekind sums is given by 1 1 h k 1 s(h, k)+ s(k, h)= + + + , −4 12 k h hk where (h, k) = 1 and h, k N := 1, 2, 3, .. -
Arxiv:2002.04395V20 [Math.GM] 3 Aug 2021
Noname manuscript No. (will be inserted by the editor) A lower bound for the modulus of the Dirichlet eta function on partition P from 2-D principal component analysis Yuri Heymann Received: date / Accepted: date Abstract The present manuscript aims to derive an expression for the lower bound of the modulus of the Dirichlet eta function on vertical lines ℜ s α. An approach based on a two-dimensional principal component analysisp match-q“ ing the dimensionality of the complex plane, which is built on a parametric ellipsoidal shape, has been undertaken to achieve this result. This lower bound, C ℜ P ?2 which is expressed as s s.t. s s , η s 1 2α , where η is the Dirichlet eta function,@ P has implicationsp q P p forq | thep q| Riemann ě | ´ hypothes| is as η s 0 for any such s P, where P is a partition spanning one half of the critical| p q| ą strip, on either sidesP of the critical line ℜ s 1 2 depending upon a variable, which regions are complementary by mirrorp q“ symmetry{ with respect to ℜ s 1 2. p q“ { Keywords Dirichlet eta function, PCA, Analytic continuation 1 Introduction The Dirichlet eta function is an alternating series related to the Riemann zeta function of interest in the field of number theory for the study of the distribu- tion of primes [12]. Both series are tied together on a two-by-two relationship 1 s expressed as η s 1 2 ´ ζ s where s is a complex number. -
Alternative Evaluation of a Ln Tan Integral Arising in Quantum Field
Alternative evaluation of a lntan integral arising in quantum field theory Mark W. Coffey Department of Physics Colorado School of Mines Golden, CO 80401 (Received 2008) November 9, 2008 Abstract A certain dilogarithmic integral I7 turns up in a number of contexts in- cluding Feynman diagram calculations, volumes of tetrahedra in hyperbolic geometry, knot theory, and conjectured relations in analytic number theory. We provide an alternative explicit evaluation of a parameterized family of inte- grals containing this particular case. By invoking the Bloch-Wigner form of the dilogarithm function, we produce an equivalent result, giving a third evaluation of I7. We also alternatively formulate some conjectures which we pose in terms of values of the specific Clausen function Cl2. arXiv:0810.5077v2 [math-ph] 15 Nov 2008 Key words and phrases Clausen function, dilogarithm function, Hurwitz zeta function, functional equation, duplication formula, triplication formula AMS classification numbers 33B30, 11M35, 11M06 1 The particular integral 24 π/2 tan t + √7 I7 ln dt, (1) ≡ 7√7 π/3 tan t √7 Z − occurs in a number of contexts and has received significant attention in the last several years [3, 4, 5, 6]. This and related integrals arise in hyperbolic geometry, knot theory, and quantum field theory [6, 7, 8]. Very recently [9] we obtained an explicit evaluation of (1) in terms of the specific Clausen function Cl2. However, much work remains. This is due to the conjectured relation between a Dirichlet L series and I7 [6], ? ∞ 1 1 1 1 1 1 I7 = L 7(2) = + + . − "(7n + 1)2 (7n + 2)2 − (7n + 3)2 (7n + 4)2 − (7n + 5)2 − (7n + 6)2 # nX=0 (2) The ? here indicates that numerical verification to high precision has been per- formed but that no proof exists, the approximate numerical value of I being I 7 7 ≃ 1.15192547054449104710169. -
On the Nontrivial Zeros of the Dirichlet Eta Function
On the nontrivial zeros of the Dirichlet eta function Vladimir Garc´ıa-Morales Departament de F´ısica de la Terra i Termodin`amica Universitat de Val`encia, E-46100 Burjassot, Spain [email protected] We construct a two-parameter complex function ηκν : C → C, κ ∈ (0, ∞), ν ∈ (0, ∞) that we call a holomorphic nonlinear embedding and that is given by a double series which is absolutely and uniformly convergent on compact sets in the entire complex plane. The function ηκν converges to the Dirichlet eta function η(s) as κ →∞. We prove the crucial property that, for sufficiently large κ, the function ∞ ηκν(s) can be expressed as a linear combination ηκν(s) = n=0 an(κ)η(s + 2νn) of horizontal shifts of the eta function (where an(κ) ∈ R and a0 = 1) and that, ∞ P indeed, we have the inverse formula η(s)= n=0 bn(κ)ηκν (s + 2νn) as well (where the coefficients b (κ) ∈ R are obtained from the a ’s recursively). By using these n P n results and the functional relationship of the eta function, η(s)= λ(s)η(1 − s), we sketch a proof of the Riemann hypothesis which, in our setting, is equivalent to the fact that the nontrivial zeros s∗ = σ∗ + it∗ of η(s) (i.e. those points for which ∗ ∗ ∗ 1 η(s )= η(1 − s ) = 0) are all located on the critical line σ = 2 . arXiv:2007.04317v1 [math.GM] 9 Jul 2020 1 1 Introduction Let s := σ + it be a complex number. The Dirichlet eta function η(s), also called alternating zeta function, is given in the half plane σ > 0 by the conditionally convergent series [1] ∞ (−1)m−1 η(s)= (1) ms m=1 X which is absolutely convergent for σ > 1. -
The Dirichlet Series That Generates the M\" Obius Function Is the Inverse Of
The Dirichlet series that generates the M¨obius function is the inverse of the Riemann zeta function in the right half of the critical strip Roupam Ghosh April 3, 2019 Abstract: In this paper I introduce a criterion for the Riemann hypothesis, and then using ∞ s 1 that I prove k=1 µ(k)/k converges for ℜ(s) > 2 . I use a step function ν(x)=2{x/2}−{Px} for the Dirichlet eta function ({x} is the fractional part of x), which was at the core of my investigations, and hence derive the stated result subsequently. arXiv:1009.1092v13 [math.GM] 18 Jan 2015 1 In 1859, Bernhard Riemann showed the existence of a deep relationship between two very different mathematical entities., viz. the zeros of an analytic function and prime numbers. The Riemann Hypothesis is usually stated as, the non-trivial zeros of the Rie- 1 mann zeta function lie on the line ℜ(s) = 2 . Although, this is the standard formulation, one of the exciting features of this problem is, it can be formulated in many different and unrelated ways. The approach I take in this paper is influenced by Beurling’s 1955 paper: A closure problem related to the Riemann zeta function and B´aez-Duarte’s 2001 paper: New versions of the Nyman-Beurling criterion for the Riemann Hypothesis, although it takes a new approach. In this paper I would be studying a simple step function ν relating it to the Dirichlet eta function η. I will show how the step function ν convolves with the M¨obius function µ(n) and gives a constant, which I think is a new result significant at attacking RH.