On the distribution of periods of holomorphic cusp forms and zeroes of period polynomials

In this paper we determine the limiting distribution of the image of the Eichler–Shimura map or equivalently the limiting joint distribution of the coefficients of the period polynomials associated to a fixed cusp form. The limiting distribution is shown to be the distribution of a certain transformation of two independent random variables both of which are equidistributed on the circle $\mathbb{R}\mathrm{/}\mathbb{Z}$, where the transformation is connected to the additive twist of the cuspidal $L$-function. Furthermore we determine the asymptotic behavior of the zeroes of the period polynomials of a fixed cusp form. We use the method of moments and the main ingredients in the proofs are additive twists of $L$-functions and bounds for both individual and sums of Kloosterman sums.