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Riemann Zeta Function zeta(2)

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The Riemann zeta function at 2 is the convergent series

zeta(2)=sum_(k=1)^infty1/(k^2)
(1)

whose value can be found using a number of different techniques (Knopp and Schur 1918, Matsuoka 1961, Stark 1969, 1970, Holme 1970, Giesy 1972, Papadimitriou 1973, Apostol 1983, Choe 1987, Kimble 1987, Yaglom and Yaglom 1987, Simmons 1992, Kortram 1996).

The finite partial sums for a positive integer n are

H_n^((2))=sum_(k=1)^(n)1/(k^2)
(2)
=zeta(2)-psi_1(n+1),
(3)

where H_n^((2)) is a generalized harmonic number (whose numerator is known as a Wolstenholme number) and psi_1(z) is the trigamma function.

The problem of finding this value analytically is sometimes known as the Basel problem (Derbyshire 2004, pp. 63 and 370) or Basler problem (Castellanos 1988). It was first proposed by Pietro Mengoli in 1644 (Derbyshire 2004, p. 370). The solution

zeta(2)=(pi^2)/6
(4)

was first found by Euler in 1735 (Derbyshire 2004, p. 64) or 1736 (Srivastava 2000).

Holme (1970), Papadimitriou (1973), and Yaglom and Yaglom (1987) all derive the result, pi^2/6 from de Moivre's identity or related identities.

zeta(2) is given by the series

zeta(2)=3sum_(k=1)^infty1/(k^2(2k; k)).
(5)

(Knopp 1990, pp. 266-267), probably known to Euler and rediscovered by Apéry.

Bailey (2000, updated 2023, pp. 10 and 24) and Borwein and Bailey (2003, pp. 128-129) give a collection of BBP-type formulas that include a number for zeta(2),

zeta(2)=3/(16)sum_(k=0)^(infty)1/(64^k)[(16)/((6k+1)^2)-(24)/((6k+2)^2)-8/((6k+3)^2)-6/((6k+4)^2)+1/((6k+5)^2)]
(6)
=1/(81)sum_(k=0)^(infty)1/(729^k)[(243)/((12k+1)^2)-(405)/((12k+2)^2)-(81)/((12k+4)^2)-(27)/((12k+5)^2)-(72)/((12k+6)^2)-9/((12k+7)^2)-9/((12k+8)^2)-5/((12k+10)^2)+1/((12k+11)^2)].
(7)

zeta(2) is given by the double series

zeta(2)=sum_(i=1)^inftysum_(j=1)^infty((i-1)!(j-1)!)/((i+j)!)
(8)

(B. Cloitre, pers. comm., Dec. 9, 2004).

The derivative of the Riemann zeta function at 2 is given by

zeta^'(2)=-sum_(k=1)^infty(lnk)/(k^2).
(9)

zeta^'(2) can be given in closed form as

zeta^'(2)=1/6pi^2[gamma+ln(2pi)-12lnA]
(10)
=-0.93754825431...
(11)

whose magnitude has decimal digits given by OEIS A073002, where gamma is the Euler-Mascheroni constant and A is the Glaisher-Kinkelin constant (given in series form by Glaisher 1894).

One derivation for zeta(2) considers the Fourier series of f(x)=x^(2n) for a nonnegative integer n on -pi<=x<=pi, extended periodically outside this interval,

f(x)=1/2a_0+sum_(m=1)^inftya_mcos(mx)+sum_(m=1)^inftyb_msin(mx),
(12)

which has coefficients for a positive integer m given by

a_0=(2pi^(2n))/(2n+1)
(13)
a_m=(2pi^(2n))/(2n+1)_1F_2(n+1/2;1/2,n+3/2;-1/4m^2pi^2)
(14)
b_m=0,
(15)

where _1F_2(a;b,c;z) is a generalized hypergeometric function and (15) is true since the integrand is odd. Therefore, the Fourier series is given explicitly by

x^(2n)=(pi^(2n))/(2n+1)+sum_(m=1)^inftya_mcos(mx).
(16)

If n=1, then

a_m=(4(-1)^m)/(m^2),
(17)

so the Fourier series is

x^2=(pi^2)/3+4sum_(m=1)^infty((-1)^mcos(mx))/(m^2).
(18)

Letting x=pi gives cos(mpi)=(-1)^m, so

pi^2=(pi^2)/3+4sum_(m=1)^infty1/(m^2),
(19)

and we have

zeta(2)=sum_(m=1)^infty1/(m^2)=(pi^2)/6.
(20)

Higher even zeta values can be obtained by finding a_m and proceeding as above.

The root linear coefficient theorem suggests another derivation, justified for the sine function by its locally uniformly convergent infinite product

sinz=zproduct_(k=1)^infty(1-(z^2)/(k^2pi^2)).
(21)

Consider the equation sinz=0 and expand sin in a Maclaurin series

sinz=z-(z^3)/(3!)+(z^5)/(5!)+...=0,
(22)

and divide by z!=0 to obtain

0=1-(z^2)/(3!)+(z^4)/(5!)+...
(23)
=1-w/(3!)+(w^2)/(5!)+...,
(24)

where w=z^2. The nonzero zeros of sinz are z=+/-kpi for positive integers k, giving w=pi^2, (2pi)^2, .... Comparing the coefficient of w in the series and the infinite product gives the sum of the reciprocals of these positive roots,

1/(pi^2)+1/((2pi)^2)+1/((3pi)^2)+...=1/(3!)=1/6,
(25)

which can be rearranged to yield

zeta(2)=(pi^2)/6.
(26)

Yet another derivation (Simmons 1992) evaluates zeta(2) using Beukers's (1979) integral

I=int_0^1int_0^1(dxdy)/(1-xy)
(27)
=int_0^1int_0^1(1+xy+x^2y^2+...)dxdy
(28)
=int_0^1[(x+1/2x^2y+1/3x^3y^2+...)]_0^1dy
(29)
=int_0^1(1+1/2y+1/3y^2+...)dy
(30)
=[y+(y^2)/(2^2)+(y^3)/(3^2)+...]_0^1
(31)
=1+1/(2^2)+1/(3^2)+...
(32)
=zeta(2).
(33)

To evaluate the integral, rotate the coordinate system by pi/4 so

x=ucostheta-vsintheta=1/2sqrt(2)(u-v)
(34)
y=usintheta+vcostheta=1/2sqrt(2)(u+v)
(35)

and

xy=1/2(u^2-v^2)
(36)
1-xy=1/2(2-u^2+v^2).
(37)

Then

I=4int_0^(sqrt(2)/2)int_0^u(dvdu)/(2-u^2+v^2)+4int_(sqrt(2)/2)^(sqrt(2))int_0^(sqrt(2)-u)(dvdu)/(2-u^2+v^2)
(38)
=I_1+I_2.
(39)

Now compute the integrals I_1 and I_2.

I_1=4int_0^(sqrt(2)/2)[int_0^u(dv)/(2-u^2+v^2)]du
(40)
=4int_0^(sqrt(2)/2)[1/(sqrt(2-u^2))tan^(-1)(v/(sqrt(2-u^2)))]_0^udu
(41)
=4int_0^(sqrt(2)/2)1/(sqrt(2-u^2))tan^(-1)(u/(sqrt(2-u^2)))du.
(42)

Make the substitution

u=sqrt(2)sintheta
(43)
sqrt(2-u^2)=sqrt(2)costheta
(44)
du=sqrt(2)costhetadtheta,
(45)

so

tan^(-1)(u/(sqrt(2-u^2)))=tan^(-1)((sqrt(2)sintheta)/(sqrt(2)costheta))=theta
(46)

and

I_1=4int_0^(pi/6)1/(sqrt(2)costheta)thetasqrt(2)costhetadtheta=(pi^2)/(18).
(47)

I_2 can also be computed analytically,

I_2=4int_(sqrt(2)/2)^(sqrt(2))[int_0^(sqrt(2)-u)(dv)/(2-u^2+v^2)]du
(48)
=4int_(sqrt(2)/2)^(sqrt(2))[1/(sqrt(2-u^2))tan^(-1)(v/(sqrt(2-u^2)))]_0^(sqrt(2)-u)du
(49)
=4int_(sqrt(2)/2)^(sqrt(2))1/(sqrt(2-u^2))tan^(-1)((sqrt(2)-u)/(sqrt(2-u^2)))du.
(50)

But

tan^(-1)((sqrt(2)-u)/(sqrt(2-u^2)))=tan^(-1)((sqrt(2)-sqrt(2)sintheta)/(sqrt(2)costheta))
(51)
=tan^(-1)((1-sintheta)/(costheta))=tan^(-1)((costheta)/(1+sintheta))
(52)
=tan^(-1)[(sin(1/2pi-theta))/(1+cos(1/2pi-theta))]
(53)
=tan^(-1){(2sin[1/2(1/2pi-theta)]cos[1/2(1/2pi-theta)])/(2cos^2[1/2(1/2pi-theta)])}
(54)
=1/2(1/2pi-theta)
(55)

for pi/6<=theta<pi/2, with the endpoint theta=pi/2 understood by a limit. Thus

I_2=4int_(pi/6)^(pi/2)1/(sqrt(2)costheta)(1/4pi-1/2theta)sqrt(2)costhetadtheta
(56)
=4[1/4pitheta-1/4theta^2]_(pi/6)^(pi/2)
(57)
=4[((pi^2)/8-(pi^2)/(16))-((pi^2)/(24)-(pi^2)/(144))]=(pi^2)/9.
(58)

Combining I_1 and I_2 gives

zeta(2)=I_1+I_2=(pi^2)/(18)+(pi^2)/9=(pi^2)/6.
(59)

See also

Apéry's Constant, Hadjicostas's Formula, Riemann Zeta Function

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References

Apostol, T. M. "A Proof That Euler Missed: Evaluating zeta(2) the Easy Way." Math. Intel. 5, 59-60, 1983.Bailey, D. H. "A Compendium of BBP-Type Formulas for Mathematical Constants." 28 Nov 2000, updated 8 Apr 2023. https://www.davidhbailey.com/dhbpapers/bbp-formulas.pdf.Beukers, F. "A Note on the Irrationality of zeta(2) and zeta(3)." Bull. London Math. Soc. 11, 268-272, 1979.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 89-90, 2003.Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988.Choe, B. R. "An Elementary Proof of sum_(n=1)^(infty)1/(n^2)=(pi^2)/6." Amer. Math. Monthly 94, 662-663, 1987.Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.Giesy, D. P. "Still Another Proof That sum1/k^2=pi^2/6." Math. Mag. 45, 148-149, 1972.Glaisher, J. W. L. "On the Constant Which Occurs in the Formula for 1^1.2^2.3^3...n^n." Messenger Math. 24, 1-16, 1894.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 37-40, 2003.Holme, F. "Ein enkel beregning av sum_(k=1)^(infty)1/(k^2)." Nordisk Mat. Tidskr. 18, 91-92 and 120, 1970.Kimble, G. "Euler's Other Proof." Math. Mag. 60, 282, 1987.Knopp, K. Theory and Application of Infinite Series. New York: Dover, 1990.Knopp, K. and Schur, I. "Über die Herleitug der Gleichung sum_(n=1)^(infty)1/(n^2)=(pi^2)/6." Archiv der Mathematik u. Physik 27, 174-176, 1918.Kortram, R. A. "Simple Proofs for sum_(k=1)^(infty)1/(k^2)=(pi^2)/6 and sinx=xproduct_(k=1)^(infty)(1-(x^2)/(k^2pi^2))." Math. Mag. 69, 122-125, 1996.Matsuoka, Y. "An Elementary Proof of the Formula sum_(k=1)^(infty)1/(k^2)=(pi^2)/6." Amer. Math. Monthly 68, 486-487, 1961.Papadimitriou, I. "A Simple Proof of the Formula sum_(k=1)^(infty)1/(k^2)=(pi^2)/6." Amer. Math. Monthly 80, 424-425, 1973.Simmons, G. F. "Euler's Formula sum_1^(infty)1/n^2=pi^2/6 by Double Integration." Ch. B. 24 in Calculus Gems: Brief Lives and Memorable Mathematics. New York: McGraw-Hill, 1992.Sloane, N. J. A. Sequence A073002 in "The On-Line Encyclopedia of Integer Sequences."Spiess, O. "Die Summe der reziproken Quadratzahlen." In Festschrift zum 60 Geburtstag von Dr. Andreas Speiser (Ed. L. V. Ahlfors et al. ). Zürich, Switzerland: Füssli, pp. 66-86, 1945.Srivastava, H. M. "Some Simple Algorithms for the Evaluations and Representations of the Riemann Zeta Function at Positive Integer Arguments." J. Math. Anal. Appl. 246, 331-351, 2000.Stark, E. L. "Another Proof of the Formula sum_(k=1)^(infty)1/(k^2)=(pi^2)/6." Amer. Math. Monthly 76, 552-553, 1969.Stark, E. L. "1-1/4+1/9-1/(16)+...=(pi^2)/(12)." Praxis Math. 12, 1-3, 1970.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 40, 1986.Yaglom, A. M. and Yaglom, I. M. Problem 145 in Challenging Mathematical Problems with Elementary Solutions, Vol. 2. New York: Dover, 1987.

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Riemann Zeta Function zeta(2)

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Weisstein, Eric W. "Riemann Zeta Function zeta(2)." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RiemannZetaFunctionZeta2.html

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