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By Waclaw Sierpiński

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Define /W = ^ C (0 ,x )-lo g r (x ). We show fi^ ) = -2 log(27 r) in the following way. (i) / " ( x ) = 0. Thus, f{x ) is of the form f{x ) = ax + 6. (ii) f { x + 1) = / ( x ) . Thus, / ( x ) = b. (iii) / ( 5 ) = Thus, /( x ) = -ilo g (2 7 r). First, we show (i).

12), and we obtain the transformation formula for rj. M e th o d 4. 3 (b)). 2. RAMANUJAN’S A AND HOLOMORPHIC EISENSTEIN SERIES 21 R emark. 5(e)). Note that, if we use the converse of the first method, we obtain a modular form theoretic proof of Euler’s pentagonal theorem. Namely, first we prove the transformation formulas for r¡ and 'd. ) by the left-hand side (77). Then using the fact that F{z) is invariant under the transformations z z+ 1 and 2: I-)“ —1 / 2:, and the fact that it is holomorphic on SL2 {Z)\H U {¿ 00}, we obtain F(z) = 1.

0 (? 4. REAL ANALYTIC EISENSTEIN SERIES For ^ Q 1 ) ’ E{s, z + 1 ) ^ ( c ^ i k^ + (c+

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