| Professor F.Smarandache[1] first gave the definition of the Smarandache ceil function in1993, i.e. for a fixed positive integer k and any positive integer n,{Sk(n)=min m∈N:n│mk}.About this function, many scholors studied its properties. Ibstedt[2] presented the following property:((?)a,b E N)(a,b)=1(?)Sk(ab)=Sk(a)Sk(b). It is easy to see that if (a, b)=1, then (Sk(a),Sk(b))=1.In her thesis, Li Jie[23]studied the Smarandache ceil function Sk(n!). If n p1α1p2α2…prαe represents the prime factorization of n, so the prime factor function Ω is defined as Ω(n)=α1+α2+…αk.She investigated the mean value of Sk(n!) by the elementary method and gave the asymptotic formula as following: where, C is a calculable constant.Feng Qiang and Guo Jinbao[24] investigated the distribution properties of the Smarandache ceil function as well as its complements Ck(n) by the analyti-cal method, where Ck(n) is defined as following:Let k≥2is a fixed integer, n is a positive integer, Ck(n)=min{m∈N:nm=tk}(t∈N).For (?)x≥1, k, n∈N, k≥2, they got the following two sharped asymptotic formulas: where In her thesis, Ren Dongmei[3] proved the asymptotic formulawhere c1and c2are computable constants, and∈is any fixed positive number. Zhang Lulu[21] proved the asymptotic formula whereP2,k(t) is a polynomial of degree2in t, δ(x)=log3/5x(log log x)-1/5, c>0is an absolute constant. The aim of this paper is to show the asymptotic formula of the function by the exponential sum method under the Riemann hypothesis We shall prove the following: Theorem1If RH is true, then we havewhere c1and c2are computable constants, and ε is any fixed positive number. In addition, we also investigate the mean value of d3(Sk(n)) under the Rie-mann hypothesis: Theorem2If RH is true, then we havewhere P2,k(t) is a polynomial of degree2in t, ε is any fixed positive number. Theorem3For1/4<θ<1/3, xθ+2e≤y≤x, we havewhere H(x)=t1xlogx+t2x, d(n) is the Dirichlet divisor function, ε is any fixed positive number, t1,t2are computable constants. |
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