Eisenstein Series

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

(1)

where the sum  excludes , , and  is an integer (Apostol 1997, p. 12).

The Eisenstein series satisfies the remarkable property

(2)

if the matrix  is in the special linear group  (Serre 1973, pp. 79 and 83). Therefore,  is a modular form of weight  (Serre 1973, p. 83).

Furthermore, each Eisenstein series is expressible as a polynomial of the elliptic invariants  and  of the Weierstrass elliptic function with positive rational coefficients (Apostol 1997).

The Eisenstein series satisfy

(3)

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

(4)

where  is a complete elliptic integral of the first kind, ,  is the elliptic modulus, and defining

(5)

we have

			(6)
			(7)

where

			(8)
			(9)
			(10)

where  is a Bernoulli number. For , 2, ..., the first few values of  are , 240, , 480, -264, , ... (OEIS A006863 and A001067).

The first few values of  are therefore

			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)

(Apostol 1997, p. 139). Ramanujan used the notations , , and , and these functions satisfy the system of differential equations

			(18)
			(19)
			(20)

(Nesterenko 1999), where  is the differential operator.

 can also be expressed in terms of complete elliptic integrals of the first kind  as

			(21)
			(22)

(Ramanujan 1913-1914), where  is the elliptic modulus. Ramanujan used the notation  and  to refer to  and , respectively.

Pretty formulas are given by

			(23)
			(24)

where  is a Jacobi theta function.

The following table gives the first few Eisenstein series  for even .

	OEIS	lattice
2	A006352
4	A004009
6	A013973
8	A008410
10	A013974

The notation  is sometimes used to refer to the closely related function

			(25)
			(26)
			(27)
			(28)
			(29)

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

(30)

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

REFERENCES:

Apostol, T. M. "The Eisenstein Series and the Invariants  and " and "The Eisenstein Series ." §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  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 ." 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. http://arxiv.org/abs/math.NT/0009130.

Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. Unpublished manuscript. 1999.

Ramanujan, S. "Modular Equations and Approximations to ." 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."