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Research Article

Open Access Peer-reviewed

F. M. S. Lima^{ }

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

In this note, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values ζ(2*k*+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(2*k*). As a result, we find an elementary proof of , as well as a recurrence formula for ζ(2*k*) from which it follows that the ratio ζ(2*k*)/π^{2}^{k} is a rational number, without making use of Euler's formula and Bernoulli numbers.

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}

(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}^{}}

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

(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

(3) |

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

(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

(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

(6) |

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

(7) |

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

and then .

Interestingly, our approach can be easily adapted to treat higher zeta values by changing the exponent of *n* from 2 to 2*k*. 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

(8) |

On rewriting the last series in terms of , one has

Now, on substituting * *in the above series, one finds

(9) |

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

(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(2*k*), 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

(11) |

which implies that .

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

(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.

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 *E*_{2m}(1) = 0 for all positive integer values of *m*.

[1] | M. Aigner and G.M. Ziegler, Proofs from THE BOOK, 5th ed., Springer, New York, 2014, Chap. 9. | ||

In article | View Article | ||

[2] | R. Ayoub, “Euler and the zeta function,” Am. Math. Monthly 81, 1067-1085 (1974). | ||

In article | View Article | ||

[3] | M. J. Dancs and T.-X. He, “An Euler-type formula for ζ(2k+1),” J. Number Theory 118, 192-199 (2006). | ||

In article | View Article | ||

[4] | L. Euler, “De summis serierum reciprocarum,” Commentarii Academiae Scientiarum Petropolitanae 7, 123-134 (1740). | ||

In article | |||

[5] | D. Kalman, “Six ways to sum a series,” Coll. Math. J. 24, 402-421 (1993). | ||

In article | View Article | ||

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

F. M. S. Lima. An Elementary Proof of and a Recurrence Formula for ζ(2*k*). *Turkish Journal of Analysis and Number Theory*. Vol. 5, No. 4, 2017, pp 143-145. http://pubs.sciepub.com/tjant/5/4/5

Lima, F. M. S.. "An Elementary Proof of and a Recurrence Formula for ζ(2*k*)." *Turkish Journal of Analysis and Number Theory* 5.4 (2017): 143-145.

Lima, F. M. S. (2017). An Elementary Proof of and a Recurrence Formula for ζ(2*k*). *Turkish Journal of Analysis and Number Theory*, *5*(4), 143-145.

Lima, F. M. S.. "An Elementary Proof of and a Recurrence Formula for ζ(2*k*)." *Turkish Journal of Analysis and Number Theory* 5, no. 4 (2017): 143-145.

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[1] | M. Aigner and G.M. Ziegler, Proofs from THE BOOK, 5th ed., Springer, New York, 2014, Chap. 9. | ||

In article | View Article | ||

[2] | R. Ayoub, “Euler and the zeta function,” Am. Math. Monthly 81, 1067-1085 (1974). | ||

In article | View Article | ||

[3] | M. J. Dancs and T.-X. He, “An Euler-type formula for ζ(2k+1),” J. Number Theory 118, 192-199 (2006). | ||

In article | View Article | ||

[4] | L. Euler, “De summis serierum reciprocarum,” Commentarii Academiae Scientiarum Petropolitanae 7, 123-134 (1740). | ||

In article | |||

[5] | D. Kalman, “Six ways to sum a series,” Coll. Math. J. 24, 402-421 (1993). | ||

In article | View Article | ||