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Eisenstein Series

An Eisenstein series with half-period ratio tau and index r is defined by

G_r(tau)=sum^'_(m=-infty)^inftysum^'_(n=-infty)^infty1/((m+ntau)^r),
(1)

where the sum sum^(') excludes m=n=0, Im[tau]>0, and r>2 is an integer (Apostol 1997, p. 12).

For integer r>=2, the Eisenstein series satisfies the transformation law

G_(2r)((atau+b)/(ctau+d))=(ctau+d)^(2r)G_(2r)(tau)
(2)

if the matrix [a b; c d] is in the special linear group SL_2(Z) (Serre 1973, pp. 79 and 83). Therefore, G_(2r) is a modular form of weight 2r (Serre 1973, p. 83).

Furthermore, each Eisenstein series G_(2k) with integer k>=2 is expressible as a polynomial of the elliptic invariants g_2 and g_3 of the Weierstrass elliptic function with positive rational coefficients (Apostol 1997).

For integer k>=2, the Eisenstein series satisfy

G_(2k)(tau)=2zeta(2k)+(2(2pii)^(2k))/((2k-1)!)sum_(n=1)^inftysigma_(2k-1)(n)e^(2piintau),
(3)

where zeta(z) is the Riemann zeta function and sigma_k(n) is the divisor function (Apostol 1997, pp. 24 and 69). Writing the nome q as

q=e^(pitaui)=e^(-piK^'(k)/K(k)),
(4)

where K(k) is a complete elliptic integral of the first kind, K^'(k)=K(sqrt(1-k^2)), k is the elliptic modulus, and defining, for integer k>=2,

E_(2k)(q)=(G_(2k)(tau))/(2zeta(2k)),
(5)

we have, for integer n>=2,

E_(2n)(q)=1+c_(2n)sum_(k=1)^(infty)(k^(2n-1)q^(2k))/(1-q^(2k))
(6)
=1+c_(2n)sum_(k=1)^(infty)sigma_(2n-1)(k)q^(2k),
(7)

where

c_(2n)=((2pii)^(2n))/((2n-1)!zeta(2n))
(8)
=(-1)^n((2pi)^(2n))/(Gamma(2n)zeta(2n))
(9)
=-(4n)/(B_(2n)),
(10)

where B_n is a Bernoulli number. For n=1, 2, ..., the first few values of c_(2n) are -24, 240, -504, 480, -264, 65520/691, ... (OEIS A006863 and A001067).

For n=1, define E_2(q) by the Fourier series below (Apostol 1997, pp. 69-71). It corresponds to the ordered lattice sum obtained by summing over m first and is not a modular form of weight 2. The first few values of E_(2n)(q) are

E_2(q)=1-24sum_(k=1)^(infty)sigma_1(k)q^(2k)
(11)
E_4(q)=1+240sum_(k=1)^(infty)sigma_3(k)q^(2k)
(12)
E_6(q)=1-504sum_(k=1)^(infty)sigma_5(k)q^(2k)
(13)
E_8(q)=1+480sum_(k=1)^(infty)sigma_7(k)q^(2k)
(14)
E_(10)(q)=1-264sum_(k=1)^(infty)sigma_9(k)q^(2k)
(15)
E_(12)(q)=1+(65520)/(691)sum_(k=1)^(infty)sigma_(11)(k)q^(2k)
(16)
E_(14)(q)=1-24sum_(k=1)^(infty)sigma_(13)(k)q^(2k),
(17)

(Apostol 1997, p. 139). Ramanujan used the notations P(z)=E_2(sqrt(z)), Q(z)=E_4(sqrt(z)), and R(z)=E_6(sqrt(z)), and these functions satisfy the system of differential equations

thetaP=1/(12)(P^2-Q)
(18)
thetaQ=1/3(PQ-R)
(19)
thetaR=1/2(PR-Q^2)
(20)

(Nesterenko 1999), where theta=zd/dz is the differential operator.

E_(2n)(q) can also be expressed in terms of complete elliptic integrals of the first kind K(k) as

E_4(q)=[(2K(k))/pi]^4(1-k^2k^('2))
(21)
E_6(q)=[(2K(k))/pi]^6(1-2k^2)(1+1/2k^2k^('2))
(22)

(Ramanujan 1913-1914), where k is the elliptic modulus. Ramanujan used the notation M(q) and N(q) to refer to E_4(q) and E_6(q), respectively.

Pretty formulas are given by

E_4(q)=1/2[theta_2^8(q)+theta_3^8(q)+theta_4^8(q)]
(23)
E_8(q)=1/2[theta_2^(16)(q)+theta_3^(16)(q)+theta_4^(16)(q)],
(24)

where theta_n(q)=theta_n(0,q) is a Jacobi theta function.

The following table gives the first few Eisenstein series E_n(q) for even n.

nOEISlatticeE_n(q)
2A0063521-24q^2-72q^4-96q^6-168q^8-...
4A004009E_81+240q^2+2160q^4+6720q^6+...
6A0139731-504q^2-16632q^4-122976q^6-...
8A008410E_8 direct sum E_81+480q^2+61920q^4+1050240q^6+...
10A0139741-264q^2-135432q^4-5196576q^6-...

The notation L(q) is sometimes used to refer to the closely related function

L(q)=1+24sum_(k=1)^(infty)sigma_1^((o))(k)(-1)^kq^k
(25)
=1-24sum_(k=1)^(infty)((2k-1)q^(2k-1))/(1+q^(2k-1))
(26)
=theta_4^4(q)-theta_2^4(q)
(27)
=[(2K(k))/pi]^2(1-2k^2)
(28)
=1-24q+24q^2-96q^3+...
(29)

(OEIS A103640), where theta_i(q) is a Jacobi theta function and

sigma_1^((o))(n)=sum_(d|n; d odd)d
(30)

is the odd divisor function (Ramanujan 2000, p. 32).

For the weight-two series, let N(Y) count the zeros rho of E_2(e^(piirho)) in the upper half-plane with 0<=Rerho<1 and Imrho>=Y, using the nome convention above. Sebbar (2026) proved that

N(Y)=1/(2piY)+O(Y^(-1/2)ln(1/Y)),
(31)

as Y->0^+, where O denotes big-O notation. Sebbar (2026) also proved that two zeros whose difference is a rational number differ by an integer, and that the values of the j-function and elliptic lambda function at a zero are transcendental. This does not unconditionally prove that the zero itself is a transcendental number.

See also

Divisor Function, Elliptic Invariants, Klein's Absolute Invariant, Leech Lattice, Pi, Theta Series, Weierstrass Elliptic Function

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References

Apostol, T. M. "The Eisenstein Series and the Invariants g_2 and g_3" and "The Eisenstein Series G_2(tau)." §1.9 and 3.10 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 12-13 and 69-71, 1997.Borcherds, R. E. "Automorphic Forms on O_(s+2,2)(R)^+ and Generalized Kac-Moody Algebras." In Proc. Internat. Congr. Math., Vol. 2. pp. 744-752, 1994.Borwein, J. M. and Borwein, P. B. "Class Number Three Ramanujan Type Series for 1/pi." J. Comput. Appl. Math. 46, 281-290, 1993.Bump, D. Automorphic Forms and Representations. Cambridge, England: Cambridge University Press, p. 29, 1997.Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 119 and 123, 1993.Coxeter, H. S. M. "Integral Cayley Numbers."The Beauty of Geometry: Twelve Essays. New York: Dover, pp. 20-39, 1999.Gunning, R. C. Lectures on Modular Forms. Princeton, NJ: Princeton Univ. Press, p. 53, 1962.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 166, 1999.Milne, S. C. "Hankel Determinants of Eisenstein Series." 13 Sep 2000. https://arxiv.org/abs/math/0009130.Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. Unpublished manuscript. 1999.Ramanujan, S. "Modular Equations and Approximations to pi." Quart. J. Pure Appl. Math. 45, 350-372, 1913-1914.Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., 2000.Sebbar, A. "Zeros of E_2, Quasi-Periods, and Schwarz Maps." 21 Oct 2026. https://arxiv.org/abs/2610.30025.Serre, J.-P. A Course in Arithmetic. New York: Springer-Verlag, 1973.Shimura, G. Euler Products and Eisenstein Series. Providence, RI: Amer. Math. Soc., 1997.Sloane, N. J. A. Sequences A001067, A004009/M5416, A006352, A006863/M5150, A008410, A013973, A013974, and A103640 in "The On-Line Encyclopedia of Integer Sequences."

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Eisenstein Series

Cite this as:

Weisstein, Eric W. "Eisenstein Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EisensteinSeries.html

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