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The Euler formula for ζ (2n)
The Riemann zeta function and Bernoulli numbers
Khristo N. Boyadzhiev
Department of Mathematics
Ohio Northern University
Ada, OH 45810
[email protected]
Abstract After a brief introduction to Bernoulli numbers we give a new proof of Euler’s
formula for the values ζ (2n) of the Riemann zeta function. The values at the negative
integers are also listed.
Contents.
1. Introduction and Bernoulli numbers
2. The computation
3. Further examples
4. The values of zeta at the negative integers
Appendix
References
1. Introduction. Bernoulli numbers
The Bernoulli numbers
appeared in the work of the Swiss mathematician Jakob
Bernoulli (1654-1705), who evaluated sums of powers of consecutive integers
.
(1.1)
The Bernoulli numbers have numerous applications in mathematics - in combinatorics, analysis,
probability. Good references for them are [7] and [12].
The exponential generating function for the Bernoulli numbers is
.
(1.2)
The relation between the two representations (1.1) and (1.2) is explained, for instance, on p. 367
in [7]. We have
, and
.
(1.3)
The very important Riemann zeta function is defined by the series
,
for
and can be extended for all
(1.4)
(see [5] on this site). Leonhard Euler (1707-1783)
first started to develop the theory of this function and evaluated it for positive even integers
. Thus Euler discovered the formula
,
(1.5)
bearing now his name. Euler’s work was published in 1755, see [9]. In particular,
.
There are many proofs of this formula, some of them elementary - see, for example, [2], [3], [4],
1
[6], [8], [11], [14], [15], [16].
Euler also tried in vain to find the exact values of zeta at the odd integers
present, we still do not know these values, we only know that
values
. At
is irrational. Finding the
is an important open problem in mathematics. In the Appendix we show part of
Euler’s original work.
The purpose of this note is to show that Euler’s formula results immediately from the
integral
(1.6)
(a Fourier sine transform, [10]) and to point out that similar Fourier transform pairs bring to
other useful relations. We shall use the well-known integral representation of the Zeta function
,
.
(1.7)
2. The computation.
The idea is simple: we expand both sides in (1.6) on powers of
. In order to expand the
integral on the left side we use the standard series
,
and for the right side we use (1.2). Comparing coefficients for
(2.1)
on both sides we obtain the
desired formula (1.5) immediately, in view of (1.7) . Some additional effort is needed, however, to
justify the substitution of (2.1) into the integral and exchanging integration and summation. This
technicality can be avoided by reasoning in the following way:
Let
to be the analytic function defined by the integral in (1.6)
.
(2.2)
2
When
, equation (1.6) provides:
,
and therefore
,
which is obviously the Maclaurin series for
, convergent for
(2.3)
in view of (1.2). The
coefficients
,
can be evaluated from (2.2) by differentiation. When n is even, they are zeroes. When
is odd, one finds:
,
(2.4)
which is (1.5).
Note that this proof can be used in reverse order to show that (1.5) leads to (1.6), and
therefore equations (1.5) and (1.6) are equivalent.
3. Further examples.
Another well-known formula, the Fourier cosine transform:
(3.1)
can also be used for a similar work. The right hand side suggests the involvement of the Euler
numbers,
, defined by the expansion:
3
,
which converges for
.(
and
for n odd, as this function is even.) At the
same time, the left side in (3.1) can be associated with the Dirichlet function:
,
(3.2)
which has integral representation
.
(3.3)
Reasoning like before, we set
.
(3.4)
Equation (3.1) shows that
.
Evaluating the coefficients
also from (3.4) one comes to the formula (see [1], [3]).
.
(3.5)
We end this section with a problem for the reader.
Problem. Find out what formula (similar to (1.5)) results from the Fourier sine transform:
.
(3.6)
4. The values of zeta at the negative integers.
4
For completeness we mention here the values of the Riemann zeta function at the negative
integers. These values can be computed by using the important functional equation satisfied by
this function
(4.1)
(see [5] on this site or [13]).
Proposition The following is true
,
(4.2)
.
Proof. If
(4.3)
is a positive odd integer, the right hand side in (4.1) is zero, as
so (4.2) is true; and if
,
, the value (4.3) follows immediately from Euler’s formula (1.5).
Appendix
Shown here is part of the original page in chapter 6 of Euler’s book [9] on the evaluation
of zeta at positive integers
5
References.
1.
M. Abramowitz, and C. A. Stegun (Eds.), Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp.
807-808, 1972
2.
Tom Apostol, Another elementary proof of Euler’s formula for
Monthly, 80 (1973), 425-432.
3.
Raymond Ayoub, Euler and the Zeta function, Amer. Math. Monthly, 81 (1974), 10671086.
4.
Bruce C. Berndt, Elementary evaluation of
154.
5.
Khristo N. Boyadzhiev, The Riemann zeta function for undergraduates, Mathematics
Bonus Files, 2007.
6.
Xuming Chen, Recursive formulas for
(1995), 372-376.
7.
Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics,:
Addison-Wesley Publ. Co., New York, 1994.
8.
John E. McCartney,
9.
Leonardo Eulero, Institutiones calculi differentialis cum ejus usu in analysi finitorum ac
doctrina serierum, Impensis academiae imperialis scientiarum Petropolitanae, 1755. Also,
another edition, Ticini: in typographeo Petri Galeatii superiorum permissu, 1787. (Opera
Omnis Ser. I (Opera Math.), Vol. X, Teubner, 1913). Online at
http://www.math.dartmouth.edu/~euler/pages/E212.html
10.
F. Oberhettinger, Tables of Fourier Transform and Fourier Transforms of
Distributions, Springer Verlag, Berlin, New York, 1990.
11.
Thomas J. Osler, Finding
(2004), 52-54.
12.
Nico M. Temme, Special Functions, An Introduction to the Classical Functions of
Mathematical Physics, John Wiley&Sons, New York, 1996.
13.
E. C. Titchmarsh and D. R. Heat-Brown, The Theory of the Riemann Zeta-Function,
2nd ed., Oxford University Press, New York 1986.
, Amer. Math.
, Math. Magazine, 48 (1975), 148-
and
, College Math. J., 26, no. 5,
from harmonic analysis, Preprint, 1992.
from a product of sines, Amer. Math. Monthly, 111
6
14.
H.Tsumura, An elementary proof of Euler’s formula for
111(5), (2004), 430-431.
15.
G. T. Williams, A new method of evaluating
19-25.
16.
Kenneth S. Williams, On
, Amer. Math. Monthly,
, Amer. Math. Monthly, 60 (1953),
, Math. Magazine, 44 (1971), 273-276.
7