Some Problems On Fourier Coefficients Of Holomorphic Cusp Forms

For a long time,the distribution of the Fourier coefficients of modular forms has attracted many attention.Ramanujan-Petersson’s conjecture predicts that for any modular form g,the order of ag(n)does not exceed nε,where ag(n)is the nth normalized Fourier coefficient of g.In 1974,for any primitive cusp form f of weight k,Deligne[7]proved the following result for k≥ 2 as a consequence of Weil’s conjecture,af(n)<<d(n)nk-1/2,where d(n)is the divisor function.In the same year,the corresponding formula for k≥1 was proved by Deligne and Serre[8].In this paper,we first consider the average behavior of the Fourier coefficients of primitive cusp forms.Assume that f is a primitive cusp form for S L(2,Z),we write λf(n)as the n-th normalized Fourier coefficient.It is an important problem to estimate the following power sum,Many researchers have paid attention to the above question when 2≤l≤8(see[9],[31],[33],[34],[38],[44],[46],etc.).In 2010,under the assumption that L(s,symlf)was automorp1ic for every l∈N,Lau and Lü[30]got an asymptotic formula of the power sum ∑n≤x λf(n)l.In 2013,Zhai[50]studied the power sum ∑a2+b2≤x λf(a2+b2)l,(2≤l≤8)at the first time.Based on the above facts,we investigate the arbitrary power sum of the Fourier coefficients of primitive cusp forms for S L(2,Z)at the point of the sum of two squares.Specifically,we have the following result.where λf(n)is the n-th normalized Fourier coefficient of f(see more detail in §1).It is a classical problem to estimate the size of sign changes of Fourier coefficients of holomorphic cusp forms.Recently,much attention is drawn to the above problem(see[21],[27],[32]etc.).In 2010,Kowalski,Lau,Soundararajan,and Wu[28]distinguished primitive cusp forms by signs in the sense of analytic density.In 2012,Matomaki[35]improved the previous result in[28]by getting more information about the behavior of the Fourier coefficients.Many authors also studied sign changes of Fourier coefficients of half integral weight cusp forms(see more details in[21],[27],[32],etc.).In 2010,Kohnen[26]obtained that there were many sign changes in the special sequence ｛af(tn2)}n∈N where f was a half integral weight cusp form and of(tn2)was real.Inspired by the above results,we consider simultaneous sign changes of Fourier coefficients of two half integral weight newforms f and g and we obtain that a certain half integral weight newform which lies in the Kohnen’s plus space can be uniquely determined by the sign of its Fourier coefficients.