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.
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 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
![]() | (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(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
![]() | (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 E2m(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 | ||
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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 | ||