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We derive new bounds for moments of the error in the Sato-Tate law overfamilies of elliptic curves. Our estimates are stronger than those obtained byW.D. Banks and I.E. Shparlinski (arXiv:math/0609144) and L. Zhao and thefist-named author in (arXiv:math/0608318) for the first and second moments, butthis comes at the cost of larger ranges of averaging. As applications, wededuce new almost-all results for the said errors and a conditional CentralLimit Theorem on the distribution of these errors. Our method is different fromthose used in the above-mentioned papers and builds on recent work by thesecond-named author and K. Sinha (arXiv:1705.04115) who derived a Central LimitTheorem on the distribution of the errors in the Sato-Tate law for families ofcusp forms for the full modular group. In addition, identities by Birch andMelzak play a crucial rule in this paper. Birch's identities connect moments ofcoefficients of Hasse-Weil $L$-functions for elliptic curves with the Kroneckerclass number and further with traces of Hecke operators. Melzak's identity iscombinatorial in nature.

Moments of the error term in the Sato-Tate law for elliptic curves