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Lambert Series and Ramanujan Proc. Indian Aead. Sei. (Math. Sci.), Vol. 103, No. 3, December 1993, pp. 269-293. Printed in India. Lambert series and Ramanujan R P AGARWAL Department of Mathematics and Astronomy, University of Lucknow, Lucknow 226007, India MS received 28 December 1992 Abstract. Lambert series are of frequent occurrence in Ramanujan's work on elliptic functions, theta functions and mock theta functions. In the present article an attempt has been made to give a critical and up-to-date account of the significant role played by Lambert series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas. Keywords. Basic hypergeometric series; Lambert series; elliptic functions; mock theta functions. 1. Introduction The topic of this article called the Lambert series is a very well-known class of series, both as analytic function and number theories. Our present interest in these series emerges out of the works of Ramanujan who has profusely and elegantly used these series in a variety of contexts in his mystic research work. A number of mathematicians, later, in an attempt to unravel the mysticism underlying some of the unproved identities of Ramanujan, have also used these series with advantage. To appreciate the effectiveness of choice of the use of these series by Ramanujan and subsequently, by other mathematicians in proving a variety of identities, one must examine how beautifully they enter into the theory of numbers, theory of Weierstrass's elliptic functions and the basic hypergeometric theory. We, now present our work starting with the definition and then the impact of Lambert series on the development of the above areas. 2. Lambert series The series a~-- (2.1) n=l 1 --X n' was considered by Lambert [19-1 in connection with the convergence of power series. oo If the series ~a~ converges, then the Lambert series (2.1) converges for all values of 1 269 270 R P Agarwal x except for x = ___ 1; otherwise it converges for those values ofx for which the series ~anx n converges. The Lambert series is used in certain problems of number theory. 1 Thus, for Ixl < 1 the sum q~(x) of the series (2.1) can be represented as a power series ~(x) = ~ %x n (2.2) 1 where a n = Eak, kin and the summation is over all divisors k of n. In particular, if a. = 1, then ~. = T(n), the number of divisors of n. The behaviour of ~b(x) (with suitable an) as xT 1 is used (see for example [20]) in the problem of Hardy and Ramanujan on obtaining an asymptotic formula for the number of unbounded partitions of a natural number. Lambert series also occur in the expansion of Eisenstein series, a particular kind of modular form. We list below a variety of familiar functions which can be expressed in terms of a Lambert series [18, pp. 448-451]: (i) If an =/~(n), where/~(n) is the M6bius function, then z = ~ p(n) 1 _ zn. m Z n z E (n) , (ii) Forlz[<l, (l-z) 2= t 1-z n where ~b(n) is the Euler function defined as the number of positive integers not exceeding n that are relatively prime to n. (iii) If ~ anzn/1 --Z n =f(z)and ~ a.z n =g(z), then one can easily see that n=l n=! f(z)= ~ O(z'). m=| (iv) For an---(- 1)n-a; an-n; an =(- 1)n-in; a.= l/n; a. =(- 1)n-l/n; a.=~"; we have for fz[ < 1, the remarkable identities, (each summation is for n -- 1 to or), Z n Z n a) E(-)"-'----1 -z" E 1 +z n Z n Z n b) = Y.O _:) z rig n Z n c) E(-)"- 1 n 1 ~-z n = E log Lambert series and Ramanujan 271 e) E (-)"- 1 Zn n 1 -z" - ~log(1 +z ~) Z" OLZ" f) ~t" i _z.- ~ 1ctz~, _ respectively. (v) In the identities (iv) (d) and (e) above, with the series of Logarithms (for which we take the principal values) one easily sees that they are equivalent to (I - z") = e -~ with co = E1 z" 1 71 1 --Z n' fi(l+z.)=e,Owithto=~(-) "-1 z" . , n 1 - z" (vi) An interesting numerical example is furnished by the Fibonacci sequence of numbers defined by Uo = 0, ui = 1 and for every n > 1, u. = u._, + u._ 2, namely 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ........ We then have ~ _ 1 =1+~+~+~-~+--+...I 1 k= * U2k ,(755 where L(x) is the sum of Lambert series F,x"/1 - x". The proof is based on the fact that u,=--, (v=O, 1,2 .... ) cc-/~ where ~t and fl are the roots of the quadratic equation x2-x - 1 =0. (vii) Lastly, an important illustration is provided by the identity O0 qN+X log F~(x) = - log(1 - q) + log q .~=o 1 q" +x' where 0 < q < 1, x > 0 and F~(x) is Jackson's q-analogue of gamma function defined as (q)~o(1 - q)l-X/(qX)~o. The integral transform F(x)=ff e =-ta(t----~)dt'l known as the Lambert transform is the continuous analogue of the Lambert series 272 R P Aoarwal (under the correspondence ta(t).-,a., e-X*--}x). The following inversion formula holds: Suppose that a(t)~L(O, oo) and that lim a(t)t 1-6 = O, 6 > O. t~+O If also z > 0 and if the function a(t) is continuous at t = z. then one has za(T)=lim(--~)k(~)k+x~lz(n)nkF'k)(n~k) k-.~ k n=l ' where ,u(n) is the M6bius function (Widder [30]). Still another case of particular interest is the Lambert series obtained by taking X n F(x) = ). a.-- n=l 1 -- X n n=l m~l = ~ bNx N, N=I where bN = ~ a,. nl,V The relation between {a, } and {b, } is equivalent to ~(s) f (s) = g(s), where f(s) and g(s) are respectively the Dirichlet series associated with {a, } and {b, }. tat) at) Hence. if f(s)= ~ a,n -s and g(s)= ~ b,n -s, then 1 1 X n n=l 1 --X n n=l if and only if ~(s) f (s) = O(s). 3. Lambert series and certain arithmetical functions We begin with an identity founded by E.T. Bell, namely q" [ 1 1 1 ] _ ~ n2 q ~ . f (1 - (3.1) n=l n~l Lambert series and Ramanujan 273 Bell obtained the formula (or its arithmetical equivalent) while looking for quadratic forms which represent all integers with at most a finite number of exceptions. An immediate equivalent of (3.1) is N.(n - s) = G(n)- Co(n), (3.2) $=1 where No(m) is the sum of the divisors of m not greater than re~a, and ~,(n) denotes the sum of rth powers of the divisors of n. Another consequence of (3.1) mentioned by Bell is that the number of representations of any integer n > 0 in the form wx + xy + yz + zu, where w, x, z, u > 0, y 1> 0, is ~2(n)--n~o(n), a classical result stated without proof by Liouville, in 1867. Bailey [12] gave three proofs for (3.1). Two of the proofs are probably as elementary as one could expect. In the first proof, he began by proving the formula z 2z 2 3z 3 z (1 - q)z 2 -- +--_-~ + + - t + 1--q 1 ~ (1-- q)(l -- z) (l -- q2)( l -- z)(1-- qz) (I -q)(l -- q2)z3 + +.-. (3.3) (1 - qa)(1 - z)(l - qz)(1 - q2z) where Izl < 1, Iql < 1. Denoting the right hand side by F(z), and writing (a), = (1 - a)(1 - aq)...(1 -- aq ~- 1), (a)o = 1, we have, on simplification F(z)-F(qz)=--- (1 -z) 2" It follows that z qz q2z F(z) =-- + -- + +... (l-z) 2 (1-qz) 2 (1-q2z)2 = ~=o q'z _ n(q'z)'= _q.' $' (1 - ~ z) ~ ,=o I1= ,=1 1 and this proves (3.3). Now differentiating (3.3) with respect to z, we obtain the formula (q),_,e' [l 1 l J .2z" + -- +... -~ = ~. --- (3.4) .=,(iLr 1-qz 1-q'-tz ,-~1 l-q" which reduces to (3.1) when z = q. 274 R P Agarwal Another result given by Bailey, in this context, was the identity a)~, (1-- aq2")q" [ 1 1 1 -- --+ + ...+ q. (1 = (1 -- qn)2(1 -- aqn) 2 L 1 q 1 - q2 ~_ a aq aq"-t ] ~ n2q, +----+ +-.--~ -aq;_ 1 = (3.5) 1-a 1--aq 1 .=1 l-q"' which reduces to Bell's identity when a = 0. Bell pointed out that if in (3.5) one expands in powers of q and equates coefficients of like powers of q, one gets a curious polynomial in a, which must vanish identically. Expressing this identically vanishing polynomial arithmetically, one can get theorems on numbers of representations and also interesting recurrence formulae for functions of divisors. By giving 'a' special values in the polynomial, for example a = - 1 or a = p, where p is a primitive rth root of unity, one obtains further theorems relating to restricted representations. In view of Bell's remarks on the number-theoretic interest of (3.5), Bailey 1-13] gave elementary algebraic proof for it. Being quite instructive we give below in some details this proof. Denote the left hand side of (3.5) by F(a).
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