The Asymptotic Distribution Of The Hybrid Arithmetic Functions

This paper studies the average order of the hybrid arithmetic function. This problem has been the concern of analytic number theory. In this paper, we studies the problem of Fourier coefficients ?f(n) of holomorphic cusp form, the sum-of-divisors function ?(n) and the Euler's function ?(n), which has some theoretical significance. Next we introduce the informations about ?f(n) and the Hecke L-function (see [7-9]).Suppose the full modular groupLet Hk* be the set of all normalized Hecke eigencuspforms of weight k for T ?SL2(Z). Then, the Fourier expansion of f ?Hk* at the cusp ? isHere, and for all positive integers m> 1, n > 1,we havewhere ?(d) is the Dirichlet character.The Hecke L-function of is defined:In 1974, P.Deligne [1] proved the Ramanujan-Petersson conjecture:|?f(n)|?d(n),where d(n) is the divisor function.Rankin [1] proved thatwhere 0 < ? < 0.06.Rankin [10] and Selberg [11] obtained the average behavior of ?f2(n) over natural numbers:In 2011,Liu, Lii, and Wu [12] studied the average behavior of ?fj(n),j =3,4,5, 6, 7, 8 over positive integer set and showed thatwhere ?j see the Theorem 1 in [12].In [3], Manski, Mayle, and Zbacnik studied the average order of da(n)?b(n)?c(n),and obtainedwhere a, b, c are real numbers, 1/2 ?ra?1, and Pn(t) is a polynomial of degree n.The aim of this paper is to research the combinations of ?fa(n),?b(n) and ?c(n),where a = 1, 2, 3,4, b, c ? R. We have the following results:Theorem 1. Let f ?Hk*,and ?f(n) denote its n-th normalized Fourier coefficient. If b, c ? R, then for any ? > 0,S1(x)(?)xb+c+1/2+?,where the implied constant depends on the cusp form f.Theorem 2. Let f?Hk* and ?f(n) denote its n-th normalized Fourier coefficient. If b, c ? R, then for any ? > 0,S2(x)= C0xb+c+1+O(xb+c+38/59+?),where the implied constant depends on the cusp form f.Theorem 3. Letf?Hk*, and ?f(n) denote its n-th normalized Fourier coefficient. If b, c ?R, then for any ? > 0,S3(x) (?) xb+c+3/4+?,where the implied constant depends on the cusp form f.Theorem 4. Let f E Hk*,and ?f(n) denote its n-th normalized Fourier coefficient. If b, c ?R, then for any ? > 0,S4(x) = xb+c+1 P1(logx) + O(xb+c+139/160+?),where P1(t) is a polynomial of degree 1, the implied constant depends on the cusp form f.