ZEROS OF EISENSTEIN SERIES

be the normalized Eisenstein series of weight k with respect to 1 := SL2(Z), where the summation extends over all coprime pairs of integers c and d and H denotes the complex upper half-plane. In [2], Rankin and Swinnerton-Dyer showed that surprisingly all the zeros of Ek in the standard fundamental domain for the action of 1 on H lie on the unit circle. This result was generalized later by Rankin [3] to certain Poincaré series and by Asai et al [1] to the function j − 744 and its images under the usual Hecke operators; here j is the classical modular invariant. The above phenomenon—as far as we can say—does not seem to be yet fully understood. In the present paper, we shall give a closed formula for the zeros of Ek on the unit circle strictly between ρ := e2πi/3 and i, in terms of an infinite series involving the Fourier coefficients of Ek , i.e. Bernoulli numbers and power divisor functions. To obtain this formula, we shall use the results of [2] and the classical Jensen formula which expresses the integral around a circle of the log modulus of a holomorphic function in terms of the log modulus of the zeros of that function lying inside the circle. Note that Jensen’s formula in connection with modular forms seems to have been used first by Rohrlich [4]. It seems possible to generalize our result to the modular functions studied in [1] and [3].