ON A FUNCTION RELATED TO RAMANUJAN ’ S TAU FUNCTION

For the function ψ=ψ12, defined by ∑1∞ψ(n)xn=x∏1∞(1−x2n)12   (|x|<1), the author derives two simple formulas. The simpler of these two formulas is expressed 
solely in terms of the well-known sum-of-divisors function.


INTRODUCTION.
Following Ramanujan [4,p. 155]we define for each positive divisor of 24 an arithmetical function as follows: (n)x n x (l-x24n/) , (I.I) an identity which is valid for each complex number x such that Ixl < I.Of course, 24 the celebrated Ramanujan tau fuction.In this paper we are specifically concerned with 12(= for simplicity).As a matter of fact, we derive two explicit formulas for .
Since these formulas involve the sum-of-divisors function and the counting function for sums of eight squares, we need the following definition.Definition.
(i) For each positive integer n, o(n) denotes the sum of all posi- tive divisors of n. (ii) for each nonnegative integer n, rk(n denotes the cardinality of the set {(Xl,X 2 Xk k In x + x +...+ x}, k an arbitrary positive integer. We can now state our main result.Theorem i.For each nonnegative integer m, m (2m+I) (1.3)In section 2 we prove theorem i, and thereafter prove a corollary which gives a formula expressing solely in terms of .
2. PROOF OF THEOREM i.Our proof requires the following three identities, each of which is valid for ach complex number uc et Ixl < I.
) is due to Euler, while (2.2) and (2.3) are due to Gauss.For proofs see [3, pp.277-284].We also need a fourth identity which the author has not been able to locate in the literature.This we here record in the following lemma.
LEMMA.For each complex number x such that Ixl < I, xm(m+l)/2}4 (2m+1)xm (2.4) 0 0 Proof: Here we need the following two identities, stated and proved in [I, p. 313]. ( We square these identities, add the resulting identities, and utilize the fact that the fourth power of the right side of (2.2) generates (-l)nr4(n), to write: In the foregoing we then let x x 2, and multiply the resulting identity by x to get (n)x n x. (l-x 2n) 12 x 2m+I (-l)ir8(i)o(2m-2i+l) 0 0 Comparing coefficients of x n we thus prove our theorem.