The Average Estimates For Fourier Coefficients Of Holomorphic Forms

According to the Langlands program,the Fourier coefficients of automorphic forms contains deep studying its various properties,will help us to solve related problems in number theory,and explore its application in the related fields.The in-formation and computing theory on SL2(Z),the Fourier coefficients of automorphic forms of information and results,which makes the Fourier coefficient theory of au-tomorphic forms has become a hot research field of number theory.Thus it is very important and essential to investigate the Fourier coefficients of automorphic forms.Let k? 1 be an even integer,and Hk*be the set of all normalized Hecke primitive eigencuspform of weight k for the full modular group SL2(Z).Each f ? Hk*has the Fourier expansion at the cusp ooThen ?f(n)is real and satisfies the multiplicative property.For any integers m ? 1 and n?1,we haveThe Hecke L-function attached to f?Hk*is defined,they can be used in the form of Euler product,where ?f(p)and ?f(p)satisfy?f(p)= ?f(p)+ ?f(p),|?f(p)| = |?f(p)| = ?f(p)?f(p)= 1.For convenience,we write ?f(p)= ?(p),?f(p)=?(p).In 1974,Deligne[1]proved the Ramanujan-Peterson conjecture|?f(n)| ? d(n),where d(n)is the Dirichlet divisor function.In 1989,Hafner and Ivic[2]obtained an ?±-results for(?).In 1990,Ivio[5]conjecturedIn 2015,Manski,Mayle,Zbacnik consider the average of d?(n)?b(n)?c(n),and provedwhere a,b,and c are real number,and Pn(t)is a polynomial in t.In Chapter 2,we investigate the average behavior of coefficient of Hecke L-function over sparse sequence by using the properties of symmetric power L-function and their Rankin-Selberg L-function.we shall consider the ?-result on error term of(?)The first aim of this paper is to give the lower bound of E(f;x).By using the Omega Theorem of Kuhleitner and Nowark(see lemma 2.1 in Section 2),we can prove the following result.Theorem 1 Let f? Hk*be a Hecke eigencuspform of even integral weight k for the full modular group,and ?f(n)denote its n-th normalized Fourier coefficient.Suppose L(sym6f × sym4f,s)and L(sym6f × sym6f,s)belong to the Selberg Class,then we havewhere c1 is a suitable constant.Theorem 2 Let f ? Hk*be a Hecke eigenc,uspform of even integral weight k for the full modular group,and ?symjf(n)denote its n-th coefficient of the sym-metric square L-function associated with f.Suppose L(sym6f × sym4f,s)and L(sym6f × sym6f,s)belong to the Selberg Class,then we havewhere c2 is a suitable constant.In Chapter 3,we estimate the sum of(?)where ?(n)is sum of divisor function,?(n)is Euler's totient function and b,c?R.By using the properties of ?(n),?(n),and ?(n),we establish the following result.Theorem 3 Let b,c ? R,then for any ?>0,we havewhere O-constant depends on f.Theorem 4 Let b,c ? R,then for any e>0,we havewhere P4(t)is a polynomial in t of degree 4 and O-constant depends on f.