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Auxiliary Polynomials in Number Theory by David Masser

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21

Modular functions

The star of this chapter is the function J, although a strong supporting role is played by j (which already had minor parts in some exercises of the previous chapter). The former can be defined by

image

which makes sense for all complex q with 0 < |q| < 1. For its properties we shall make constant reference to Lang (1973), here abbreviated to L, as well as Rankin (1977), abbreviated to R. It is analytic there (L pp. 39, 44, 249; or R p. 199), but its Laurent expansion

image (21.1)

(L p. 45 or R p. 199) is somewhat less illuminating ...

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