Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

An Elementary Proof of and a Recurrence Formula for ζ(2k)

F. M. S. Lima

Institute of Physics, University of Brasília, P.O. Box 04455, 70919-970, Brasília-DF, Brazil

Abstract: In this note, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values ζ(2k+1), ζ(s) being the Riemann zeta function and k a positive integer, is modified in a manner to furnish the even zeta values ζ(2k). As a result, we find an elementary proof of , as well as a recurrence formula for ζ(2k) from which it follows that the ratio ζ(2k)/π²k is a rational number, without making use of Euler's formula and Bernoulli numbers.

Keywords: Riemann zeta function, Euler’s formula, Euler polynomials

Received May 27, 2017; Revised June 27, 2017; Accepted June 27, 2017

Citation: F. M. S. Lima, An Elementary Proof of and a Recurrence Formula for ζ(2k). Turkish Journal of Analysis and Number Theory, 5 (4). 143-145. January 2017. doi:10.12691/tjant-5-4-5

1. Introduction

For real values of , , the Riemann zeta function is defined as . In this domain, this series converges according to the integral test.{1} For , , , Euler (1740) did find that [4]

[picture] | (1)

where is the k-th Bernoulli number, i.e. the rational coefficient of in the Taylor series expansion of , . As a consequence, since one has , which is the Euler solution to the Basel problem (see Ref. [2] and references therein).

In Ref. [3], Dancs and He (2006) introduced a series expansion approach to derive Euler-type formulae for . On noting that their approach could be modified in a manner to furnish similar formulas for , here in this note we show that the change of by in the Dancs-He initial series in fact yields a series expansion which can be reduced to a finite sum involving only even zeta values. From the first few terms of this sum, we have found an elementary proof of and a recurrence formula for . The proofs are elementary in the sense they do not involve complex analysis, Fourier series, or multiple integrals.{²}

2. Elementary Evaluation of ζ(2)

For any real and , we begin by taking into account the following Taylor series expansion considered by Dancs and He in Ref. [3]:

[picture] | (2)

which converges absolutely for .

From the generating function for the Euler polynomial , i.e. , it is clear that , for all nonnegative integer values of m. For , we have

[picture] | (3)

Let us take this series as our definition of , being a positive integer. Therefore

[picture] | (4)

for all integer .

Now, let

be an auxiliary function, with u belonging to the same domain as above. Since , then can be written in the form

On expanding in a Taylor series, one finds

in which the change of sums justifies by Fubini's theorem. By writing the last series in terms of , one has

[picture] | (5)

This is enough for the derivation of our first result.

Theorem 1 (Short evaluation of ζ(2)). .

Proof. By taking the limit as on both sides of Eq.(5), one has

[picture] | (6)

which, in face of the value of stated in Eq.(4), implies that

[picture] | (7)

Since and for all , the right-hand side of this equation reduces to ,{³} which implies that

and then .

3. Recurrence Formula for ζ(2k)

Interestingly, our approach can be easily adapted to treat higher zeta values by changing the exponent of n from 2 to 2k. The result is the following recurrence formula for even zeta values.

Theorem 2 (Recurrence for ζ(2k)). For any positive integer k,

Proof. We begin by defining . Again, since , we may write

[picture] | (8)

On rewriting the last series in terms of , one has

Now, on substituting in the above series, one finds

[picture] | (9)

The limit as , taken on both sides of Eq.(9), yields

[picture] | (10)

From Eq.(4), one knows that

For nonnegative values of m, one has , the only exception being . This reduces Eq.(10) to

By extracting the last term of the sum and isolating ζ(2k), one finds

A multiplication by 2 on both sides yields the desired result.

The first few even zeta values can be readily obtained from the recurrence formula in Theorem 2. For , the sum in the right-hand side is null and our recurrence reduces to

which simplifies to , in agreement to our Theorem 1. For , our recurrence yields

By substituting and multiplying both sides by 4, one finds

[picture] | (11)

which implies that .

Note that, by writing the recurrence formula in Theorem 2 in the form

[picture] | (12)

it is straightforward to show, by induction on k, that the ratio is a rational number for every positive integer k, without making use of Euler's formula for , as stated in Eq.(1), and Bernoulli numbers. In fact, this was the original motivation that has led the author to study the properties of the Dancs-He series expansions. The proofs developed here could well be modified to cover other special functions of interest in analytic number theory.

Notes

1. For s=1, one has the harmonic series , which diverges to infinity.

2. For non-elementary proofs, see, e.g., Refs. [1,5] and references therein.

3. This occurs because E2m(1) = 0 for all positive integer values of m.

References

[1]   | M. Aigner and G.M. Ziegler, Proofs from THE BOOK, 5th ed., Springer, New York, 2014, Chap. 9.
[2]   | R. Ayoub, “Euler and the zeta function,” Am. Math. Monthly 81, 1067-1085 (1974).
[3]   | M. J. Dancs and T.-X. He, “An Euler-type formula for ζ(2k+1),” J. Number Theory 118, 192-199 (2006).
[4]   | L. Euler, “De summis serierum reciprocarum,” Commentarii Academiae Scientiarum Petropolitanae 7, 123-134 (1740).
[5]   | D. Kalman, “Six ways to sum a series,” Coll. Math. J. 24, 402-421 (1993).