Formulas for Odd Zeta Values and Powers of π
|
|
- Dominick Harmon
- 5 years ago
- Views:
Transcription
1 Journal of Integer Sequences, Vol. 4 (0), Article..5 Formulas for Odd Zeta Values and Powers of π Marc Chamberland and Patrick Lopatto Department of Mathematics and Statistics Grinnell College Grinnell, IA 50 USA chamberl@math.grinnell.edu Abstract Plouffe conjectured fast converging series formulas for π n+ and ζ(n+) for small values of n. We find the general pattern for all integer values of n and offer a proof. Introduction It took nearly one hundred years for the Basel Problem finding a closed form solution to k= /k to see a solution. Euler solved this in 735 and essentially solved the problem where the power of two is replaced with any even power. This formula is now usually written as ζ(n) = ( ) n+b n(π) n, (n)! where ζ(s) is the Riemann zeta function and B k is the k th Bernoulli number uniquely defined by the generating function x e x = n=0 B n x n, x < π. n! and whose first few values are 0, /, /6, 0, /30,... However, finding a closed form for ζ(n + ) has remained an open problem. Only in 979 did Apéry show that ζ(3) is irrational. His proof involved the snappy acceleration ( ) n ζ(3) = 5 n= n 3( n n ).
2 This tidy formula does not generalize to the other odd zeta values, but other representations, such as nested sums or integrals, have been well-studied. The hunt for a clean result like Euler s has largely been abandoned, leaving researchers with the goal of finding formulas which either converge quickly or have an elegant form. Following his success in discovering a new formula for π, Simon Plouffe[3] postulated several identities which relate either π m or ζ(m) to three infinite series. Letting S n (r) = k= k n (e πrk ), the first few examples are and π = 7S () 96S () + 4S (4) π 3 = 70S 3 () 900S 3 () + 80S 3 (4) π 5 = 7056S 5 () 6993S 5 () + 63S 5 (4) π 7 = S 7 () 70875S 7 () S 5(4) ζ(3) = 8S 3 () 37S 3 () + 7S 3 (4) ζ(5) = 4S 5 () 59 0 S 5() + 0 S 5(4) ζ(7) = S 7() 03 4 S 7() S 7(4). Plouffe conjectured these formulas by first assuming, for example, that there exist constants a, b, and c such that π = as () + bs () + cs (4). By obtaining accurate approximations of each the three series, he wrote some computer code to postulate rational values for a, b, c. Today, such integer relations algorithms have been used to discover many formulas. The widely used PSLQ algorithm, developed by Ferguson and Bailey[], is implemented in Maple. The following Maple code (using Maple 4) solves the above problem: > with(integerrelations): > Digits := 00; > S := r -> sum( /k/( exp(pi*r*k)- ), k=..infinity ); > PSLQ( [ Pi, S(), S(), S(4) ] ); The PSLQ command returns the vector [, 7, 96, 4], producing the first formula. While the computer can be used to conjecture the coefficients of a specific power, finding the general sequence of rationals has remained an open problem. This note finds the sequences and offers formal proofs.
3 Exact Formulas While it does not seem that ζ(n+) is a rational multiple of π n+, a result in Ramanujan s notebooks gives a relationship with infinite series which converge quickly. Theorem. (Ramanujan) If α > 0, β > 0, and αβ = π, then { } α n ζ(n + ) + S n+(α) = { } ( β) n ζ(n + ) + S n+ n+(β) 4 n ( ) k B k B n+ k (k)!(n + k)! αn+ k β k. Using α = β = π in Proposition and defining we have n+ F n = ( ) k B k B n+ k (k)!(n + k)!, ( π n ( π) n) ( ) ζ(n + ) + S n+(π) = 4 n π n+ F n. To find formulas for the odd zeta values and powers of π, we will divide these into two classes: ζ(4m ) and ζ(4m + ). Such distinctions can be seen in other studies; see [, pp ]. First we find the formulas for π 4m and ζ(4m ). If n is odd, then ζ(n + ) + S n+(π) = 4n πn+ F n () Using α = π/ and β = π in Proposition and defining one has n+ G n = ( 4) k B k B n+ k (k)!(n + k)!, ζ(n + ) = S n+(4π) + 4 n S n+ (π) + 4n (4 n + ) Combining this with equation () yields πn+ G n. 4 n S n+ (π) (4 n + )S n+ (π) + S n+ (4π) 4 n (4n + )F n 4n G n = π n+ Substituting n = m and defining = 4 m [(4 m + )F m G m ]/ produces 3
4 π 4m = 4m S 4m (π) 4m + S 4m (π) + S 4m (4π) This identity may be used in conjunction with equation () to obtain ζ(4m ) = F m 4 4m S 4m (π) + G m 4 m S 4m (π) F m 4 m S 4m (4π). To obtain formulas for the 4m + cases, use α = π/ and β = π in Theorem with n = m to obtain 4 m ζ(4m + ) = π4m+ G m S 4m+ (4π) + 4 m S 4m+ (π) (. () 4m ) Define T n (r) (similar to S n (r)) by T n (r) = k= k n (e rk + ) and another finite sum of Bernoulli numbers by H n = n ( 4) n+k B 4k B 4n+ 4k (4k)!(4n + 4k)!. Vepstas [4] cites an expression credited to Ramanujan: ( + ( 4) m 4m+ )ζ(4m + ) = T 4m+ (π) + ( 4m+ ( 4) m )S 4m+ (π) + 4m+ π 4m+ H m + 4m π 4m+ G m. Vepstas also produces a formula to show the relationship between T k and S k : T k (x) = S k (s) S k (x) Combining the last two equations produces ( ) + ( 4) m 4m+ 4 m ( 4m ) π4m+ G m S 4m+ (4π) + 4 m S 4m+ (π) = [ 4m+ ( 4) m + ]S 4m+ (π) 4S 4m+ (4π) + 4m+ π 4m+ H m + 4m π 4m+ G m Letting and K m = ( 4m ) + ( 4) m 4m+ E m = 4m G m 4m+ K m H m 4m K m G m, 4
5 one eventually finds π 4m+ = 4m S 4m+ (π) + K m[ 4m+ ( 4) m + ] S 4m+ (π) + ( 4K m) S 4m+ (4π) E m E m Substituting this into equation () produces E m References ζ(4m + ) = 6m (G(m)6 m + E m ) ( + 6 m )E m S 4m+ (π) G mk m 6 m ( 6 m ( 4) m + ) ( + 6 m )E m S 4m+ (π) G(m)6m + 4G m 6 m K m + E m ( + 6 m )E m S 4m+ (4π). [] D. Bailey and J. Borwein. Experimentation in Mathematics. AK Peters, 004. [] H. R. P. Ferguson and D. H. Bailey. A polynomial time, numerically stable integer relation algorithm. RNR Techn. Rept. RNR-9-03, Jul. 4, 99. [3] Plouffe, S. Identities inspired by Ramanujan notebooks (part ), April 006. [4] Vepstas, L. On Plouffe s Ramanujan identities, November Mathematics Subject Classification: Primary Y60. Keywords: π, Riemann zeta function. Received January 4 0; revised version received February 7 0. Published in Journal of Integer Sequences, February 0 0. Return to Journal of Integer Sequences home page. 5
The Value of the Zeta Function at an Odd Argument
International Journal of Mathematics and Computer Science, 4(009), no., 0 The Value of the Zeta Function at an Odd Argument M CS Badih Ghusayni Department of Mathematics Faculty of Science- Lebanese University
More informationAn Euler-Type Formula for ζ(2k + 1)
An Euler-Type Formula for ζ(k + ) Michael J. Dancs and Tian-Xiao He Department of Mathematics and Computer Science Illinois Wesleyan University Bloomington, IL 670-900, USA Draft, June 30th, 004 Abstract
More informationFormulae for some classical constants
Formulae for some classical constants Alexandru Lupaş (to appear in Proceedings of ROGER-000 The goal of this paper is to present formulas for Apéry Constant, Archimede s Constant, Logarithm Constant,
More informationBinary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes
3 47 6 3 Journal of Integer Sequences, Vol. 6 003), Article 03.3.7 Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes Marc Chamberland Department of Mathematics and Computer Science
More informationBINARY BBP-FORMULAE FOR LOGARITHMS AND GENERALIZED GAUSSIAN-MERSENNE PRIMES. Marc Chamberland
1 BINARY BBP-FORMULAE FOR LOGARITHMS AND GENERALIZED GAUSSIAN-MERSENNE PRIMES Marc Chamberland Department of Mathematics and Computer Science Grinnell College Grinnell, IA, 5011 E-mail: chamberl@math.grinnell.edu
More informationInteger Relation Methods : An Introduction
Integer Relation Methods : An Introduction Special Session on SCIENTIFIC COMPUTING: July 9 th 2009 Jonathan Borwein, FRSC www.carma.newcastle.edu.au/~jb616 Laureate Professor University of Newcastle, NSW
More informationIdentities Inspired by the Ramanujan Notebooks Second Series
Identities Inspired by the Ramanujan Notebooks Second Series by Simon Plouffe First draft August 2006 Revised March 4, 20 Abstract A series of formula is presented that are all inspired by the Ramanujan
More informationBernoulli Numbers and their Applications
Bernoulli Numbers and their Applications James B Silva Abstract The Bernoulli numbers are a set of numbers that were discovered by Jacob Bernoulli (654-75). This set of numbers holds a deep relationship
More informationAnalytic Aspects of the Riemann Zeta and Multiple Zeta Values
Analytic Aspects of the Riemann Zeta and Multiple Zeta Values Cezar Lupu Department of Mathematics University of Pittsburgh Pittsburgh, PA, USA PhD Thesis Overview, October 26th, 207, Pittsburgh, PA Outline
More informationTHE MEAN-MEDIAN MAP MARC CHAMBERLAND AND MARIO MARTELLI
THE MEAN-MEDIAN MAP MARC CHAMBERLAND AND MARIO MARTELLI 1. Introduction In the last few decades mathematicians have discovered very simple functions which induce spectacularly intricate dynamics. For example,
More informationThe Riemann and Hurwitz zeta functions, Apery s constant and new rational series representations involving ζ(2k)
The Riemann and Hurwitz zeta functions, Apery s constant and new rational series representations involving ζ(k) Cezar Lupu 1 1 Department of Mathematics University of Pittsburgh Pittsburgh, PA, USA Algebra,
More informationMathematics 324 Riemann Zeta Function August 5, 2005
Mathematics 324 Riemann Zeta Function August 5, 25 In this note we give an introduction to the Riemann zeta function, which connects the ideas of real analysis with the arithmetic of the integers. Define
More informationMultiple Zeta Values of Even Arguments
Michael E. Hoffman U. S. Naval Academy Seminar arithmetische Geometrie und Zahlentheorie Universität Hamburg 13 June 2012 1 2 3 4 5 6 Values The multiple zeta values (MZVs) are defined by ζ(i 1,..., i
More informationBeukers integrals and Apéry s recurrences
2 3 47 6 23 Journal of Integer Sequences, Vol. 8 (25), Article 5.. Beukers integrals and Apéry s recurrences Lalit Jain Faculty of Mathematics University of Waterloo Waterloo, Ontario N2L 3G CANADA lkjain@uwaterloo.ca
More informationEvaluating ζ(2m) via Telescoping Sums.
/25 Evaluating ζ(2m) via Telescoping Sums. Brian Sittinger CSU Channel Islands February 205 2/25 Outline 2 3 4 5 3/25 Basel Problem Find the exact value of n= n 2 = 2 + 2 2 + 3 2 +. 3/25 Basel Problem
More informationConstruction Of Binomial Sums For π And Polylogarithmic Constants Inspired By BBP Formulas
Applied Mathematics E-Notes, 77, 7-46 c ISSN 67-5 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Construction Of Binomial Sums For π And Polylogarithmic Constants Inspired By BBP
More informationSOME FORMULAS OF RAMANUJAN INVOLVING BESSEL FUNCTIONS. Henri Cohen
SOME FORMULAS OF RAMANUJAN INVOLVING BESSEL FUNCTIONS by Henri Cohen Abstract. We generalize a number of summation formulas involving the K-Bessel function, due to Ramanujan, Koshliakov, and others. Résumé.
More information2 BORWEIN & BRADLEY Koecher's formula points up a potential problem with symbolic searching. Namely, negative results need to be interpreted carefully
SEARCHING SYMBOLICALLY FOR APERY-LIKE FORMULAE FOR VALUES OF THE RIEMANN ZETA FUNCTION Jonathan Borwein and David Bradley Abstract. We discuss some aspects of the search for identities using computer algebra
More informationInteger Powers of Arcsin
Integer Powers of Arcsin Jonathan M. Borwein and Marc Chamberland February 5, 7 Abstract: New simple nested sum representations for powers of the arcsin function are given. This generalization of Ramanujan
More informationExperimental Determination of Apéry-like Identities for ζ(2n +2)
Experimental Determination of Apéry-lie Identities for ζ(n + David H. Bailey, Jonathan M. Borwein, and David M. Bradley CONTENTS. Introduction. Discovering Theorem. 3. Proof of Theorem.. An Identity for
More informationA gentle introduction to PSLQ
A gentle introduction to PSLQ Armin Straub Email: math@arminstraub.com April 0, 200 Abstract This is work in progress. Please let me know about any comments and suggestions. What PSLQ is about PSLQ is
More informationπ-day, 2013 Michael Kozdron
π-day, 2013 Michael Kozdron What is π? In any circle, the ratio of the circumference to the diameter is constant. We are taught in high school that this number is called π. That is, for any circle. π =
More informationA NOTE ON RECURRENCE FORMULA FOR VALUES OF THE EULER ZETA FUNCTIONS ζ E (2n) AT POSITIVE INTEGERS. 1. Introduction
Bull. Korean Math. Soc. 5 (4), No. 5,. 45 43 htt://dx.doi.org/.434/bkms.4.5.5.45 A NOTE ON RECURRENCE FORMULA FOR VALUES OF THE EULER ZETA FUNCTIONS ζ E (n) AT POSITIVE INTEGERS Hui Young Lee and Cheon
More informationAlgorithms for Experimental Mathematics I David H Bailey Lawrence Berkeley National Lab
Algorithms for Experimental Mathematics I David H Bailey Lawrence Berkeley National Lab All truths are easy to understand once they are discovered; the point is to discover them. Galileo Galilei Algorithms
More informationA New Kind of Science: Ten Years Later
A New Kind of Science: Ten Years Later David H. Bailey May 7, 202 It has been ten years since Stephen Wolfram published his magnum opus A New Kind of Science [4]. It is worth re-examining the book and
More informationarxiv: v3 [math.nt] 9 May 2011
ON HARMONIC SUMS AND ALTERNATING EULER SUMS arxiv:02.592v3 [math.nt] 9 May 20 ZHONG-HUA LI Department of Mathematics, Tongji University, No. 239 Siping Road, Shanghai 200092, China Graduate School of Mathematical
More informationAutomatic Sequences and Transcendence of Real Numbers
Automatic Sequences and Transcendence of Real Numbers Wu Guohua School of Physical and Mathematical Sciences Nanyang Technological University Sendai Logic School, Tohoku University 28 Jan, 2016 Numbers
More informationFIFTH ROOTS OF FIBONACCI FRACTIONS. Christopher P. French Grinnell College, Grinnell, IA (Submitted June 2004-Final Revision September 2004)
Christopher P. French Grinnell College, Grinnell, IA 0112 (Submitted June 2004-Final Revision September 2004) ABSTRACT We prove that when n is odd, the continued fraction expansion of Fn+ begins with a
More informationComputing Bernoulli Numbers Quickly
Computing Bernoulli Numbers Quickly Kevin J. McGown December 8, 2005 The Bernoulli numbers are defined via the coefficients of the power series expansion of t/(e t ). Namely, for integers m 0 we define
More informationThe Miraculous Bailey-Borwein-Plouffe Pi Algorithm
Overview: 10/1/95 The Miraculous Bailey-Borwein-Plouffe Pi Algorithm Steven Finch, Research and Development Team, MathSoft, Inc. David Bailey, Peter Borwein and Simon Plouffe have recently computed the
More informationFamilies of Solutions of a Cubic Diophantine Equation. Marc Chamberland
1 Families of Solutions of a Cubic Diophantine Equation Marc Chamberland Department of Mathematics and Computer Science Grinnell College, Grinnell, IA, 50112, U.S.A. E-mail: chamberl@math.grin.edu. This
More informationOld and new algorithms for computing Bernoulli numbers
Old and new algorithms for computing Bernoulli numbers University of New South Wales 25th September 2012, University of Ballarat Bernoulli numbers Rational numbers B 0, B 1,... defined by: x e x 1 = n
More informationOn a series of Ramanujan
On a series of Ramanujan Olivier Oloa To cite this version: Olivier Oloa. On a series of Ramanujan. Gems in Experimental Mathematics, pp.35-3,, . HAL Id: hal-55866 https://hal.archives-ouvertes.fr/hal-55866
More informationElementary Analysis on Ramanujan s Nested Radicals
A project under INSPIRE-SHE program. Elementary Analysis on Ramanujan s Nested Radicals Gaurav Tiwari Reg. No. : 6/009 1 Contents Contents... 1 Abstract... About Ramanujan... 3 Introduction to Nested Radicals...
More informationarxiv: v2 [math.nt] 19 Apr 2017
Evaluation of Log-tangent Integrals by series involving ζn + BY Lahoucine Elaissaoui And Zine El Abidine Guennoun arxiv:6.74v [math.nt] 9 Apr 7 Mohammed V University in Rabat Faculty of Sciences Department
More informationarxiv: v2 [math.nt] 28 Feb 2010
arxiv:002.47v2 [math.nt] 28 Feb 200 Two arguments that the nontrivial zeros of the Riemann zeta function are irrational Marek Wolf e-mail:mwolf@ift.uni.wroc.pl Abstract We have used the first 2600 nontrivial
More informationExtracting Hexadecimal Digits of π
Extracting Hexadecimal Digits of π William Malone William Malone is currently a senior Mathematics major at Ball State University. Upon graduation he plans to pursue graduate studies starting in the fall
More informationIt's the last class. Who cares about the syllabus?
It's the last class (and it's sunny out) Who cares about the syllabus? CS 2800: Discrete Structures, Spring 2015 Sid Chaudhuri hdwallpapers.cat https://youtu.be/ofjgpgpq0na Persi Diaconis, Professor of
More informationComputer discovery of new mathematical facts and formulas
Computer discovery of new mathematical facts and formulas David H. Bailey http://www.davidhbailey.com Lawrence Berkeley National Laboratory (retired) University of California, Davis, Department of Computer
More informationRamanujan and Euler s Constant
Richard P. Brent MSI, ANU 8 July 2010 In memory of Ed McMillan 1907 1991 Presented at the CARMA Workshop on Exploratory Experimentation and Computation in Number Theory, Newcastle, Australia, 7 9 July
More informationMathematics 805 Homework 9 Due Friday, April 3, 1 PM. j r. = (et 1)e tx t. e t 1. = t n! = xn 1. = x n 1
Mathematics 85 Homework 9 Due Friday, April 3, PM. As before, let B n be the Bernoulli polynomial of degree n. a Show that B n B n n n. b By using part a, derive a formula epressing in terms of B r. Answer:
More informationThe Bernoulli Numbers. Kyler Siegel Stanford SUMO April 23rd, 2015
The Bernoulli Numbers Kyler Siegel Stanford SUMO April 23rd, 2015 Recall the formulas See https://gowers.wordpress.com/2014/11/04/sums-of-kth-powers/ See https://gowers.wordpress.com/2014/11/04/sums-of-kth-powers/
More informationRiemann Hypotheses. Alex J. Best 4/2/2014. WMS Talks
Riemann Hypotheses Alex J. Best WMS Talks 4/2/2014 In this talk: 1 Introduction 2 The original hypothesis 3 Zeta functions for graphs 4 More assorted zetas 5 Back to number theory 6 Conclusion The Riemann
More informationSome Interesting Properties of the Riemann Zeta Function
Some Interesting Properties of the Riemann Zeta Function arxiv:822574v [mathho] 2 Dec 28 Contents Johar M Ashfaque Introduction 2 The Euler Product Formula for ζ(s) 2 3 The Bernoulli Numbers 4 3 Relationship
More informationInverse Symbolic Calculation: Jonathan Borwein, FRSC Computer Assisted Research Mathematics and Applications CARMA
Inverse Symbolic Calculation: symbols from numbers Jonathan Borwein, FRSC www.carma.newcastle.edu.au/~jb616 Laureate Professor University of Newcastle, NSW Director, Centre for Computer Assisted Research
More informationInteger Relation Detection and Lattice Reduction David H. Bailey 1. To appear in Computing in Science and Engineering
Integer Relation Detection and Lattice Reduction David H. Bailey To appear in Computing in Science and Engineering. Introduction Let x =(x,x,,x n ) be a vector of real or complex numbers. x is said to
More informationVerbatim copying and redistribution of this document. are permitted in any medium provided this notice and. the copyright notice are preserved.
New Results on Primes from an Old Proof of Euler s by Charles W. Neville CWN Research 55 Maplewood Ave. West Hartford, CT 06119, U.S.A. cwneville@cwnresearch.com September 25, 2002 Revision 1, April, 2003
More informationHypergeometric series and the Riemann zeta function
ACTA ARITHMETICA LXXXII.2 (997) Hypergeometric series and the Riemann zeta function by Wenchang Chu (Roma) For infinite series related to the Riemann zeta function, De Doelder [4] established numerous
More informationIntroduction to Algebraic Number Theory Part I
Introduction to Algebraic Number Theory Part I A. S. Mosunov University of Waterloo Math Circles November 7th, 2018 Goals Explore the area of mathematics called Algebraic Number Theory. Specifically, we
More information1. Introduction This paper investigates the properties of Ramanujan polynomials, which, for each k 0, the authors of [2] define to be
ZEROS OF RAMANUJAN POLYNOMIALS M. RAM MURTY, CHRIS SMYTH, AND ROB J. WANG Abstract. In this paper, we investigate the properties of Ramanujan polynomials, a family of reciprocal polynomials with real coefficients
More informationIrrationality proofs, moduli spaces, and dinner parties
Irrationality proofs, moduli spaces, and dinner parties Francis Brown, IHÉS-CNRS las, Members Colloquium 17th October 2014 1 / 32 Part I History 2 / 32 Zeta values and Euler s theorem Recall the Riemann
More informationElementary Analysis on Ramanujan s Nested Radicals
A project under INSPIRE-SHE program. Elementary Analysis on Ramanujan s Nested Radicals by Gaurav Tiwari Reg. No. : 6/009 1 CONTENTS Abstract... About Ramanujan...3 Definitions... 4 Denesting... 5 How
More informationINFORMATION-THEORETIC EQUIVALENT OF RIEMANN HYPOTHESIS
INFORMATION-THEORETIC EQUIVALENT OF RIEMANN HYPOTHESIS K. K. NAMBIAR ABSTRACT. Riemann Hypothesis is viewed as a statement about the capacity of a communication channel as defined by Shannon. 1. Introduction
More informationYour quiz in recitation on Tuesday will cover 3.1: Arguments and inference. Your also have an online quiz, covering 3.1, due by 11:59 p.m., Tuesday.
Friday, February 15 Today we will begin Course Notes 3.2: Methods of Proof. Your quiz in recitation on Tuesday will cover 3.1: Arguments and inference. Your also have an online quiz, covering 3.1, due
More informationSOME FAMOUS UNSOLVED PROBLEMS. November 18, / 5
SOME FAMOUS UNSOLVED PROBLEMS November 18, 2014 1 / 5 SOME FAMOUS UNSOLVED PROBLEMS Goldbach conjecture: Every even number greater than four is the sum of two primes. November 18, 2014 1 / 5 SOME FAMOUS
More information#A08 INTEGERS 9 (2009), ON ASYMPTOTIC CONSTANTS RELATED TO PRODUCTS OF BERNOULLI NUMBERS AND FACTORIALS
#A08 INTEGERS 9 009), 83-06 ON ASYMPTOTIC CONSTANTS RELATED TO PRODUCTS OF BERNOULLI NUMBERS AND FACTORIALS Bernd C. Kellner Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen,
More informationNew Multiple Harmonic Sum Identities
New Multiple Harmonic Sum Identities Helmut Prodinger Department of Mathematics University of Stellenbosch 760 Stellenbosch, South Africa hproding@sun.ac.za Roberto Tauraso Dipartimento di Matematica Università
More informationII Computer-assisted Discovery and Proof
II Computer-assisted Discovery and Proof Seminar Australian National University (November 14, 2008) Jonathan Borwein, FRSC www.cs.dal.ca/~jborwein Canada Research Chair in Collaborative Technology Laureate
More informationResearch Article Operator Representation of Fermi-Dirac and Bose-Einstein Integral Functions with Applications
Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2007, Article ID 80515, 9 pages doi:10.1155/2007/80515 Research Article Operator Representation of Fermi-Dirac
More informationGoldbach and twin prime conjectures implied by strong Goldbach number sequence
Goldbach and twin prime conjectures implied by strong Goldbach number sequence Pingyuan Zhou E-mail:zhoupingyuan49@hotmail.com Abstract In this note, we present a new and direct approach to prove the Goldbach
More informationSOME INEQUALITIES FOR THE q-digamma FUNCTION
Volume 10 (009), Issue 1, Article 1, 8 pp SOME INEQUALITIES FOR THE -DIGAMMA FUNCTION TOUFIK MANSOUR AND ARMEND SH SHABANI DEPARTMENT OF MATHEMATICS UNIVERSITY OF HAIFA 31905 HAIFA, ISRAEL toufik@mathhaifaacil
More informationarxiv: v5 [math.gm] 1 Jun 2018
The Product eπ Is Irrational N A Carella arxiv:70608394v5 [mathgm] Jun 208 Abstract: This note shows that the product eπ of the natural base e and the circle number π is an irrational number Contents The
More informationLucas Congruences. A(n) = Pure Mathematics Seminar University of South Alabama. Armin Straub Oct 16, 2015 University of South Alabama.
Pure Mathematics Seminar University of South Alabama Oct 6, 205 University of South Alabama A(n) = n k=0 ( ) n 2 ( n + k k k ) 2, 5, 73, 445, 3300, 89005, 2460825,... Arian Daneshvar Amita Malik Zhefan
More informationIntroduction on Bernoulli s numbers
Introduction on Bernoulli s numbers Pascal Sebah and Xavier Gourdon numbers.computation.free.fr/constants/constants.html June 2, 2002 Abstract This essay is a general and elementary overview of some of
More informationNanjing Univ. J. Math. Biquarterly 32(2015), no. 2, NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS
Naning Univ. J. Math. Biquarterly 2205, no. 2, 89 28. NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS ZHI-WEI SUN Astract. Dirichlet s L-functions are natural extensions of the Riemann zeta function.
More informationTHE ARITHMETIC OF THE COEFFICIENTS OF HALF INTEGRAL WEIGHT EISENSTEIN SERIES. H(1, n)q n =
THE ARITHMETIC OF THE COEFFICIENTS OF HALF INTEGRAL WEIGHT EISENSTEIN SERIES ANTAL BALOG, WILLIAM J. MCGRAW AND KEN ONO 1. Introduction and Statement of Results If H( n) denotes the Hurwitz-Kronecer class
More informationConvergence of Some Divergent Series!
Convergence of Some Divergent Series! T. Muthukumar tmk@iitk.ac.in 9 Jun 04 The topic of this article, the idea of attaching a finite value to divergent series, is no longer a purely mathematical exercise.
More informationComputing Bernoulli numbers
Computing Bernoulli numbers David Harvey (joint work with Edgar Costa) University of New South Wales 27th September 2017 Jonathan Borwein Commemorative Conference Noah s on the Beach, Newcastle, Australia
More informationApproximation of π by Numerical Methods Mathematics Coursework (NM)
Approximation of π by Numerical Methods Mathematics Coursework (NM) Alvin Šipraga Magdalen College School, Brackley April 1 1 Introduction There exist many ways to approximate 1 π, usually employed in
More informationrama.tex; 21/03/2011; 0:37; p.1
rama.tex; /03/0; 0:37; p. Multiple Gamma Function and Its Application to Computation of Series and Products V. S. Adamchik Department of Computer Science, Carnegie Mellon University, Pittsburgh, USA Abstract.
More informationRunning Modulus Recursions
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 13 (2010), Article 10.1.6 Running Modulus Recursions Bruce Dearden and Jerry Metzger University of North Dakota Department of Mathematics Witmer Hall
More informationIDENTITIES ABOUT INFINITE SERIES CONTAINING HYPERBOLIC FUNCTIONS AND TRIGONOMETRIC FUNCTIONS. Sung-Geun Lim
Korean J. Math. 19 (2011), No. 4, pp. 465 480 IDENTITIES ABOUT INFINITE SERIES CONTAINING HYPERBOLIC FUNCTIONS AND TRIGONOMETRIC FUNCTIONS Sung-Geun Lim Abstract. B. C. Berndt established many identities
More informationMATH 271 Summer 2016 Practice problem solutions Week 1
Part I MATH 271 Summer 2016 Practice problem solutions Week 1 For each of the following statements, determine whether the statement is true or false. Prove the true statements. For the false statement,
More informationRelations for Nielsen polylogarithms
Relations for Nielsen polylogarithms (Dedicated to Richard Askey at 80 Jonathan M. Borwein and Armin Straub April 5, 201 Abstract Polylogarithms appear in many diverse fields of mathematics. Herein, we
More informationResearch Article Integer Powers of Arcsin
Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 007 Article ID 938 0 pages doi:0.55/007/938 Research Article Integer Powers of Arcsin Jonathan M. Borwein
More informationHigh Performance Computing Meets Experimental Mathematics
High Performance Computing Meets Experimental Mathematics David H. Bailey Lawrence Berkeley National Laboratory, USA David Broadhurst Department of Physics, Open University, UK Yozo Hida University of
More informationNumerical evaluation of the Riemann Zeta-function
Numbers, constants and computation Numerical evaluation of the Riemann Zeta-function Xavier Gourdon and Pascal Sebah July 23, 2003 We expose techniques that permit to approximate the Riemann Zeta function
More informationOn a secant Dirichlet series and Eichler integrals of Eisenstein series
On a secant Dirichlet series and Eichler integrals of Eisenstein series Oberseminar Zahlentheorie Universität zu Köln University of Illinois at Urbana Champaign November 12, 2013 & Max-Planck-Institut
More informationBernoulli Polynomials
Chapter 4 Bernoulli Polynomials 4. Bernoulli Numbers The generating function for the Bernoulli numbers is x e x = n= B n n! xn. (4.) That is, we are to expand the left-hand side of this equation in powers
More informationOn Some Combinations of Non-Consecutive Terms of a Recurrence Sequence
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 21 (2018), Article 18.3.5 On Some Combinations of Non-Consecutive Terms of a Recurrence Sequence Eva Trojovská Department of Mathematics Faculty of Science
More informationPart II. Number Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler
More information6 Lecture 6b: the Euler Maclaurin formula
Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Fall 217 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 217 March 26, 218 6 Lecture 6b: the Euler Maclaurin formula
More informationA Simple Proof that ζ(n 2) is Irrational
A Simple Proof that ζ(n 2) is Irrational Timothy W. Jones February 3, 208 Abstract We prove that partial sums of ζ(n) = z n are not given by any single decimal in a number base given by a denominator of
More informationGoldbach Conjecture: An invitation to Number Theory by R. Balasubramanian Institute of Mathematical Sciences, Chennai
Goldbach Conjecture: An invitation to Number Theory by R. Balasubramanian Institute of Mathematical Sciences, Chennai balu@imsc.res.in R. Balasubramanian (IMSc.) Goldbach Conjecture 1 / 22 Goldbach Conjecture:
More informationFor many years, researchers have dreamt INTEGER RELATION DETECTION T HEME ARTICLE. the Top
the Top T HEME ARTICLE INTEGER RELATION DETECTION Practical algorithms for integer relation detection have become a staple in the emerging discipline of experimental mathematics using modern computer technology
More informationON EVALUATION OF RIEMANN ZETA FUNCTION ζ(s) 1. Introduction. It is well-known that the Riemann Zeta function defined by. ζ(s) = n s, R(s) > 1 (2)
ON EVALUATION OF RIEMANN ZETA FUNCTION ζ(s) QIU-MING LUO, BAI-NI GUO, AND FENG QI Abstract. In this paper, by using Fourier series theory, several summing formulae for Riemann Zeta function ζ(s) and Dirichlet
More informationParametric Euler Sum Identities
Parametric Euler Sum Identities David Borwein, Jonathan M. Borwein, and David M. Bradley September 23, 2004 Introduction A somewhat unlikely-looking identity is n n nn x m m x n n 2 n x, valid for all
More informationExperimental mathematics and integration
Experimental mathematics and integration David H. Bailey http://www.davidhbailey.com Lawrence Berkeley National Laboratory (retired) Computer Science Department, University of California, Davis October
More informationOn Exponentially Perfect Numbers Relatively Prime to 15
On Exponentially Perfect Numbers Relatively Prime to 15 by Joseph F. Kolenick Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Mathematics Program YOUNGSTOWN
More informationExistence of a Limit on a Dense Set, and. Construction of Continuous Functions on Special Sets
Existence of a Limit on a Dense Set, and Construction of Continuous Functions on Special Sets REU 2012 Recap: Definitions Definition Given a real-valued function f, the limit of f exists at a point c R
More informationApplications of Fourier Series and Zeta Functions to Genocchi Polynomials
Appl. Math. Inf. Sci., No. 5, 95-955 (8) 95 Applied Mathematics & Information Sciences An International Journal http://d.doi.org/.8576/amis/58 Applications of Fourier Series Zeta Functions to Genocchi
More informationarxiv: v6 [math.nt] 12 Sep 2017
Counterexamples to the conjectured transcendence of /α) k, its closed-form summation and extensions to polygamma functions and zeta series arxiv:09.44v6 [math.nt] Sep 07 F. M. S. Lima Institute of Physics,
More information(1) = 0 = 0,
GALLERY OF WALKS ON THE SQUARE LATTICE BY A TURING PLOTTER FOR BINARY SEQUENCES RICHARD J. MATHAR Abstract. We illustrate infinite walks on the square lattice by interpretation of binary sequences of zeros
More informationON THE TAYLOR COEFFICIENTS OF THE HURWITZ ZETA FUNCTION
ON THE TAYLOR COEFFICIENTS OF THE HURWITZ ZETA FUNCTION Khristo N. Boyadzhiev Department of Mathematics, Ohio Northern University, Ada, Ohio, 45810 k-boyadzhiev@onu.edu Abstract. We find a representation
More informationA class of Dirichlet series integrals
A class of Dirichlet series integrals J.M. Borwein July 3, 4 Abstract. We extend a recent Monthly problem to analyse a broad class of Dirichlet series, and illustrate the result in action. In [5] the following
More informationEuler-Maclaurin summation formula
Physics 4 Spring 6 Euler-Maclaurin summation formula Lecture notes by M. G. Rozman Last modified: March 9, 6 Euler-Maclaurin summation formula gives an estimation of the sum N in f i) in terms of the integral
More informationProblem Set 5 Solutions
Problem Set 5 Solutions Section 4.. Use mathematical induction to prove each of the following: a) For each natural number n with n, n > + n. Let P n) be the statement n > + n. The base case, P ), is true
More informationRelating log-tangent integrals with the Riemann zeta function arxiv: v1 [math.nt] 17 May 2018
Relating log-tangent integrals with the Riemann zeta function arxiv:85.683v [math.nt] 7 May 8 Lahoucine Elaissaoui Zine El-Abidine Guennoun May 8, 8 Abstract We show that integrals involving log-tangent
More informationarxiv: v1 [math.gm] 11 Jun 2012
AM Comp. Sys. -9 Author version Dynamical Sieve of Eratosthenes Research arxiv:20.279v math.gm] Jun 202 Luis A. Mateos AM Computer Systems Research, 070 Vienna, Austria Abstract: In this document, prime
More informationRAMANUJAN: A TALE OF TWO EVALUATIONS
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 46, Number 3, 06 RAMANUJAN: A TALE OF TWO EVALUATIONS DONALD J. MANZOLI ABSTRACT. In 887, beneath a canopy of stars, Srinivasa Ramanujan commenced his brief
More information