Eisenstein Series

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


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

The Eisenstein series satisfies the remarkable property


if the matrix [a b; c d] is in the special linear group SL_n(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 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).

The Eisenstein series satisfy


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


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


we have




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).

The first few values of E_(2n)(q) are therefore


(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


(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


(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


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.

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


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

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

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

See also

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

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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., 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.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, A006863/M5150, A008410, A013973, A013974, and A103640 in "The On-Line Encyclopedia of Integer Sequences."

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Eisenstein Series

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Weisstein, Eric W. "Eisenstein Series." From MathWorld--A Wolfram Web Resource.

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