ON THE INTEGER PART OF A POSITIVE INTEGER S K-TH ROOT

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1 ON THE INTEGER PART OF A POSITIVE INTEGER S K-TH ROOT Yang Hai Research Center for Basic Science, Xi an Jiaotong University, Xi an, Shaanxi, P.R.China Fu Ruiqin School of Science, Xi an Shiyou University, Xi an, Shaanxi, P.R.China Abstract Keywords: The ain urose of this aer is using the eleentary ethod and analytic ethod to study the asytotic roerties of the integer art of the k-th root ositive integer, and give two interesting asytotic forulae. k-th root; Integer art; Asytotic forula.. Introduction And Results For any ositive integer n, let sn) denote the integer art of k-th root of n. For exale, s) =, s) =, s3) =, s4) =,, s k ) =, s k + ) =,, s3 k ) =,. In roble 80 of [], Professor F.Sarandache asked us to study the roerties of the sequence sn). About this roble, it sees that none had studied it, at least we have not seen related aer before. In this aer, we use the eleentary ethod and analytic ethod to study the asytotic roerties of this sequence, and obtain two interesting asytotic forulae. That is, we shall rove the following: Theore. For any real nuber x >, we have the asytotic forula Ωsn)) = x ln ln x + A log k) x + O x ln x where Ωn) denotes the total nuber of rie divisors of n, A is a constant. Theore. Let be a fixed ositive integer and ϕn) be the Euler totient function, then for any real nuber x, we have the asytotic forula ϕsn), )) = h)x + k + )h) + O x +ε) k, where sn), ) denotes the greatest coon divisor of sn) and, h) = + α α ), and ε is any ositive nuber. ϕ) α ),

2 6 SCIENTIA AGNA VOL., NO.. Soe Leas To colete the roof of the theores, we need the following two sile leas. Lea. For any real nuber x >, then we have Ωn) = x log log x + Ax + O x log x where A = γ + log ) + ) + ), γ is the Euler constant. Proof. See reference[]. Lea. Let be a fixed ositive integer and ϕn) be the Euler totient function, then for any real nuber x, we have the asytotic forula ϕ, n)) = x h) + O x +ε), where, n) denotes the greatest coon divisor of and n, h) = + α α ), and ε is any ositive nuber. ϕ) α Proof. Let α F s) = n= ϕ, n)) n s, then fro the Euler Product forula [3] and the ultilicative roerty of ϕ, n), we ay get F s) = ϕ, )) + s + ϕ, ) )) s + = ϕ, )) + s + + ϕ, α )) α )s + ϕ, ) α )) αs ) α s + s + + ) s = ζs) ϕ, )) + s + + ϕ, ) α )) α )s ) s + ϕ, ) α )) αs, where ζs) is the Rieann zeta-function. Obviously, we have inequality ϕ, n)) < K, n= ϕ, n)) n σ < ), K σ, where σ > is the real art of s. So by Perron forula [4], taking b =, T = x, Hx) = K, Bσ) =, then we have K σ ϕ, n)) = πi +it it ζs)rs) xs s ds + O x +ε),

3 On the integer art of a ositive integer s k-th root 63 where Rs) = α + To estiate the ain ter ϕ, )) s + + ϕ, ) α )) α )s ) s + ϕ, ) α )) αs. +it ζs)rs) xs πi it s ds we ove the integral line fro s = ± it to s = / ± it. This tie, we have a sile ole oint at s = with residue R)x. That is +it + +it + πi it +it it + +it ) it it Taking T = x 3/, we can easily get the estiate and πi it Noting that R) = ζs)rs) xs ds = R)x. s +it ) it + ζs)rs) xs πi +it it s ds x ζσ + it )Rs)x dσ T T = x, ζs)rs) xs +it s ds α = ϕ) + ϕ, )) + α α ). α So we have the asyotic forula where h) = ϕ). T 0 ζ + it)rs)x t dt x +ε. + + ϕ, α )) α ) ϕ, n)) = x h) + O x +ε), α ) ) + ϕ, ) α )) α + α α ). This coletes the roof of Lea

4 64 SCIENTIA AGNA VOL., NO. 3. Proof of Theores In this section, we will colete the roof of Theores. First we rove Theore. For any real nuber x >, let be a fixed ositive integer with k x + ) k, fro the definition of sn) we have Ωsn)) = = = = k t= t= t= t ) k n<t k Ωsn)) + t= t k n<t+) k Ωsn)) + [t + ) k t k ]Ωt) + O k n<x ) t k Ωt) + O k log, Ωsn)) Ω) k Ω) k n<+) k where we have used the estiate Ω) log n. Let By) = Ωn), then by Able s identity and Lea, we can easily deduce that n y t k Ωt) = k B) k ) y k By)dy t= = k log log + A) k ) y k log log y + Ay k )dy ) k +O log = k log log + A k k ) k k log log + A k k ) + O log = k k log log + ) k k A + O. log Therefore, we can obtain the asytotic forula ) Ωsn)) = k k log log + A + O. log On the other hand, we also have the estiate 0 x k < + ) k k x k k.

5 On the integer art of a ositive integer s k-th root 65 Now cobining the above, we ay iediately obtain the asytotic forula ) x Ωsn)) = x log log x + A log k) x + O. log x This coletes the roof of Theore. Now we coe to rove Theore. For any fixed ositive integer, we have ϕsn), )) = ϕ[n k ], )) = k ], )) + + k ], )) + O N ε ) i< k ϕ[i N i<n+) k ϕ[i = [j + ) k j k ]ϕj, )) + O N ε ). j N Fro Lea, we can let AN) = ϕj, )) = N h) + O N +ε), j N fj) = [j + ) k j k ], Then by Able s identity, we can easily obtain [j + ) k j k ]ϕj, )) j N N = AN)fN) A)f) At)f t)dt = [N h) + O N +ε) ][N + ) k N k ] N [t h) + O t +ε) ] k[t + ) k t k ]dt = k N k h) + O k N k +ε) k )h)n k ) k N k +ε) = k + N k + )h) + O = h)x + k + )h) + +O This coletes the roof of Theore. References x k +ε). [] F. Sarandache, Only Probles, Not Solutions, Xiquan Publishing House, Chicago, 993. [] G. H. Hardy and S. Raanujian, The noral nuber of rie factors of a nuber n, Quart. J. ath ), 76-9.

6 66 SCIENTIA AGNA VOL., NO. [3] To. Aostol, Introduction to Analytic Nuber Theory, New York, Sringer-Verlag, 976. [4] Pan Chengdong and Pan Chengbiao, Foundation of Analytic nuber Theory, Beijing, Science Press, 997.