Re: [其他] 數論證明題

[Sfly (topos)]

※ 引述《JohnMash (Paul)》之銘言：
: 令
: Λ_k(n) = Σ_{d|n} μ(d) ( ㏒(n/d))^k
: 試證
: 當 n 的質因數個數大於k時 Λ_k(n) = 0

let L represent the function log
then, by definite, Λ_k = μ*(L^k), where * means convolution.

Clearly, the Mangoldt function Λ=Λ_1 = μ*L
                                 =L．(μ*1) - (μL)*1
                                 = (-μL)*1

Now, use (log(n/d))^k = (log(n/d)^{k-1})(log n - log d),
we see that Λ_k = L．Λ_{k-1} + (-μL)*L^{k-1}
                 = L．Λ_{k-1} + (μ*Λ)*L^{k-1}
                 = L．Λ_{k-1} + Λ*Λ_{k-1}.

Hence the result.

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