Some Nonlinear Identities for Divisor Functions
In his 1916 paper, Ramanujan derived a series of nonlinear identities for a class of divisor functions. We now know that these identities arise as a consequence of the low dimension of certain spaces of modular forms. In this chapter, we explain these ide
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Some Nonlinear Identities for Divisor Functions
1 A Quadratic Relation Amongst Divisor Functions Let σs (n) denote the sum of the sth powers of the divisors of n. Also, set 1 σs (0) = ζ (−s) 2 where as usual ζ (s) denotes the Riemann zeta function. In his fundamental paper “On Certain Arithmetical Functions”, Ramanujan showed that there is a quadratic relation involving σs (n). We have σr (j ) = dr . j ≤x
j ≤x dt=j
Interchanging the summation, the right-hand side is seen to be d r [x/d] = x d r−1 + O d r = O x r+1 . d≤x
d≤x
d≤x
On the other hand, we could also write this as 1 (x/t)r+1 . dr ∼ r + 1 t≤x t≤x d≤x/t
The right-hand side is seen to be 1 x r+1 ζ (r + 1). r +1 The same method will allow us to treat σr (j )(x − j )s ∼
j ≤x
and for integers x = n, we find that it is Γ (r + 1)Γ (s + 1) ∼ ζ (r + 1)nr+s+1 Γ (r + s + 2) provided that r > 0 and s ≥ 0. M.R. Murty, V.K. Murty, The Mathematical Legacy of Srinivasa Ramanujan, DOI 10.1007/978-81-322-0770-2_8, © Springer India 2013
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Some Nonlinear Identities for Divisor Functions
Now let us set Σr,s (n) = σr (0)σs (n) + · · · + σr (n)σs (0). Then, using the fact that
ns < σs (n) < ns 1−s + 2−s + · · · = ns ζ (s)
it follows that for r > 0 and s ≥ 0, lim inf
Σr,s (n) Γ (r + 1)Γ (s + 1) ζ (r + 1) ≥ r+s+1 Γ (r + s + 2) n
and for r > 0 and s > 1, lim sup
Σr,s (n) Γ (r + 1)Γ (s + 1) ζ (r + 1)ζ (s). ≤ Γ (r + s + 2) nr+s+1
Ramanujan proves that whenever r and s are positive odd integers, Γ (r + 1)Γ (s + 1) ζ (r + 1)ζ (s + 1) σr+s+1 (n) Γ (r + s + 2) ζ (r + s + 2) 2 ζ (1 − r) + ζ (1 − s) + nσr+s−1 (n) + O n 3 (r+s+1) . r +s Moreover, he shows that in a finite number of cases, there is no error term. In other words, the O-term above can be removed, and one has an exact identity. The values of (r, s) for which this occurs are given in the table below. Σr,s (n) =
r
s
1 1 1 1 1 3 3 3 5
1 3 5 7 11 3 5 9 7
In particular, the case r = 1 and s = 3 gives the identity σ1 (1)σ3 (n) + σ1 (3)σ3 (n − 1) + σ1 (5)σ3 (n − 2) + · · · + σ1 (2n + 1)σ3 (0) 1 σ5 (2n + 1). = 240
2 Quadratic Relations Amongst Eisenstein Series Since the Fourier coefficients of Eisenstein series are given in terms of the functions σk (n), the relations of the previous section may be interpreted as quadratic relations satisfied by Eisenstein series. Using the fact that the space of modular forms of level
3 A Formula for the τ -Function
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1 and weight k is one-dimensional for 4 ≤ k ≤ 10 and for k = 14, one gets the following relations: E8 (z) = E4 (z)2 , E10 (z) = E4 (z)E6 (z), 428000 E12 (z) − E8 (z)E4 (z) = − (z) 691 and 762048 (z). 691 All of these identities can be expressed as relations between the functions σk (n) for various values of k and τ (n). Ghate [59] has classified all monomial relations between Eisenstein series. E12 (z) − E6 (z)2 =
3 A Formula for the τ -Function Building on the ideas of the previous section, we can also ask for relations between Eisenstein series and cusp forms. There is, of course, the relation 1728(z) = E4
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