A remark on Zagier's observation of the Mandelbrot set

We investigate the arithmetic properties of the coefficients for the normalized conformal mapping of the exterior of the Multibrot set. In this paper, an estimate of the prime factor of the denominator of these coefficients is given. Bielefeld, Fisher and Haeseler presented Zagier's observation for the growth of denominator of the coefficients for the Mandelbrot set and Yamashita verified it in his master thesis. Our study takes into consideration both Zagier's observation and Yamashita's estimate.