On the coefficients in the expansions of certain modular functions

1. A very large proportion of the most interesting arithmetical functions —of the functions, for example, which occur in the theory of partitions, the theory of the divisors of numbers, or the theory of the representation of numbers by sums of squares—occur as the coefficients in the expansions of elliptic modular functions in powers of the variableq =e πi τ. All of these functions have a restricted region of. existence, the unit circle |q | = 1 being a “ natural boundary” or line of essential singularities. The most important of them, such as the functions (ω1 /π)12 ∆ =q 2 {(1-q 2 ) (1-q 4 )...}24 , (1, 1) ϑ3 (0) = 1 + 2q + 2q 4 + 2q 9 + , (1. 2) 12 (ω1 /π)4 g 2 = 1 + 240 (13 q 2 /1-q 2 + 23 q 4 /1-q 4 + ...), (1, 3) 216 (ω1 /π)6 g 3 = 1 - 504 (15 q 2 /1-q 2 + 25 q 4 /1-q 4 + ...), (1, 4) are regular inside the unit circle ; and many, such as the functions (1, 1) and (1, 2), have the additional property of having no zeros inside the circle, so that their reciprocals are also regular. In a series of recent papers we have applied a new method to the study of these arithmetical functions. Our aim has been to express them as series which exhibit explicitly their order of magnitude, and the genesis of their irregular variations asn increases. We find, for example, forp (n ) the number of unrestricted partitions ofn ,and forr s (n ), the number of representations ofn as the sum of an even numbers of squares, the series