Equidistribution of Fourier coefficients of square and cube symmetric \(L\)-functions
The main theorems of this work are three equidistribution results about the Fourier coefficients of square and cube symmetric \(L\)-functions of primitive cusp forms. The first provides a new proof of an equidistribution result for the Fourier coefficients of symmetric square \(L\)-functions implied by the work of \textit{P. Sarnak} [Analytic number theory and diophantine problems, Proc. Conf., Stillwater/Okla. 1984, Prog. Math. 70, 321--331 (1987; Zbl 0628.10028)], \textit{J.-P. Serre} [J. Am. Math. Soc. 10, No.1, 75--102 (1997; Zbl 0871.11032)] and \textit{J. B. Conrey, W. Duke} and \textit{D. W. Farmer} [Acta Arith. 78, No. 4, 405--409 (1997; Zbl 0876.11020)]. The new proof does not use their work. The analogue for symmetric cube \(L\)-cubes is the content of the second main theorem of the paper. In the final theorem, it is proved that the Fourier coefficients of symmetric square \(L\)-functions are ``harmonically equidistributed'' according to the definition given in the paper.
scientific article; zbMATH DE number 5633534