OFFSET
1,2
COMMENTS
Number 10 of the 74 eta-quotients listed in Table I of Martin (1996). Cusp form level 4 weight 5.
Called chi_4(n) by Glaisher and Hardy because as Glaisher (1907) writes on page 21 "It can be shown (see section 53) that chi_4(n) admits of an arithmetical definition, being in fact equal to one-fourth of the sum of the fourth powers of all complex numbers which have n as norm, viz. chi_4(n) = 1/4 sum_n (a + i b)^4, where a + i b is any number which has n for norm. It is in consequence of this definition that the notation chi_4(n) has been used." - Michael Somos, Jun 18 2012
REFERENCES
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135 section 9.3. MR0106147 (21 #4881)
H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 175, 4.7 Exercise 5. MR1471703 (98g:14032)
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 34).
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Case k=10, called F(tau).
M. Rogers, J.G. Wan, and I. J. Zucker, Moments of Elliptic Integrals and Critical L-Values, arXiv:1303.2259 [math.NT], 2013.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of phi(q)^2 * psi(-q)^8 = chi(q)^6 * psi(-q)^10 = f(q)^3 * psi(-q)^7 = f(-q^2)^6 * psi(-q)^4 = f(-q^2)^10 / chi(q)^4 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Mar 12 2013
Expansion of eta(q)^4 * eta(q^2)^2 * eta(q^4)^4 in powers of q.
G.f.: x * (Product_{k>0} (1 - x^k)^4 * (1 - x^(2*k))^2 * (1 - x^(4*k))^4).
G.f.: (t*t'' - 3(t')^2) / 2 where t = theta_3(x) (A000122) and t' := x * (dt/dx), t'' := (t')'. - Michael Somos, Nov 08 2005
Euler transform of period 4 sequence [ -4, -6, -4, -10, ...]. - Michael Somos, Jul 17 2004
a(n) is multiplicative with a(2^e) = (-4)^e, a(p^e) = p^(2*e) * (1 + (-1)^e)/2 if p == 3 (mod 4), a(p^e) = a(p) * a(p^(e-1)) - p^4 * a(p^(e-2)) for p == 1 (mod 4) where a(p) = 2 * Re( (x + i*y)^4 ) and p = x^2 + y^2 with even x. - Michael Somos, Nov 18 2014
Given A = A0 + A1 + A2 + A3 is the 4-section, then 0 = (A0^2 - A2^2)^2 + 4 * A0*A2*A1^2. - Michael Somos, Mar 08 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 32 (t/i)^5 f(t) where q = exp(2 Pi i t). - Michael Somos, May 28 2013
a(4*n + 3) = 0. - Michael Somos, Mar 12 2013
a(2*n) = -4 * a(n). a(4*n + 1) = A215472(n). - Michael Somos, Sep 05 2013
a(n) = 1/4 * Sum_{a^2 + b^2 = n} (a + bi)^4 = Sum_{a > 0, b >= 0, a^2 + b^2 = n} (a + bi)^4. - Seiichi Manyama, Apr 25 2017
EXAMPLE
G.f. = q - 4*q^2 + 16*q^4 - 14*q^5 - 64*q^8 + 81*q^9 + 56*q^10 - 238*q^13 + ...
From Seiichi Manyama, Apr 25 2017: (Start)
a(1) = (1 + 0i)^4 = 1,
a(2) = (1 + 1i)^4 = -4,
a(4) = (2 + 0i)^4 = 16,
a(5) = (1 + 2i)^4 + (2 + 1i)^4 = -7 - 24i - 7 + 24i = -14,
a(8) = (2 + 2i)^4 = -64,
a(9) = (3 + 0i)^4 = 81,
a(10) = (1 + 3i)^4 + (3 + 1i)^4 = 28 - 96i + 28 + 96i = 56 (End)
MATHEMATICA
If[SquaresR[2, #]==0, 0, 1/4 Plus@@((x+I y)^4/.{ToRules[Reduce[x^2+y^2==#, {x, y}, Integers]]})] &/@Range[70] (* Ant King, Nov 10 2012 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^2)^2, {q, 0, n}]; (* Michael Somos, May 28 2013 *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A))^4 * eta(x^2 + A)^2, n))}; /* Michael Somos, Jul 17 2004 */
(PARI) {a(n) = local(r); if( n<1, 0, r = sqrtint(n); sum( x=-r, r, sum( y=-r, r, if( x^2 + y^2 == n, (x + I*y)^4) )) / 4 )}; /* Michael Somos, Sep 12 2005 */
(PARI) {a(n) = my(A, p, e, x, y, z, a0, a1); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (-4)^e, p%4 == 3, if( e%2, 0, p^(2*e)), forstep( i=0, sqrtint(p), 2, if( issquare( p - i^2, &y), x = i; break)); a0 = 1; a1 = x = real( (x + I*y)^4 ) * 2; for( i=2, e, y = x*a1 - p^4*a0; a0=a1; a1=y); a1))) }; /* Michael Somos, Nov 18 2014 */
(Sage) CuspForms( Gamma1(4), 5, prec=71).0; # Michael Somos, May 28 2013
(Magma) Basis( CuspForms( Gamma1(4), 5), 71) [1]; /* Michael Somos, May 27 2014 */
CROSSREFS
KEYWORD
sign,mult
AUTHOR
STATUS
approved