Compact form of $pi_k(x)$

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Let, $pi_k(x)$ denote the number of $nle x$ with $k$-prime factors (not necessarily distinct). Using the Sieve of Eratostheness, show that, $$pi_k(x) le fracx(Alog log x+B)^kk! log x.$$ for some positive constants $A$ and $B$.



Im unable to start the problem, as I can not write $pi_k(x)$ is in compact form. If a number $N$ has $k$-prime factors (not necessarily distinct) means if $N=p_1^r_1p_2^r_2cdots p_t^r_t$ then $r_1+r_2+cdots +r_t=k$. So how can I write $pi_k(x)$ in compact form ? Any hint.?










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  • maybe $pi(x^1/k)^k$
    – Kenta S
    Sep 6 at 10:19














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Let, $pi_k(x)$ denote the number of $nle x$ with $k$-prime factors (not necessarily distinct). Using the Sieve of Eratostheness, show that, $$pi_k(x) le fracx(Alog log x+B)^kk! log x.$$ for some positive constants $A$ and $B$.



Im unable to start the problem, as I can not write $pi_k(x)$ is in compact form. If a number $N$ has $k$-prime factors (not necessarily distinct) means if $N=p_1^r_1p_2^r_2cdots p_t^r_t$ then $r_1+r_2+cdots +r_t=k$. So how can I write $pi_k(x)$ in compact form ? Any hint.?










share|cite|improve this question





















  • maybe $pi(x^1/k)^k$
    – Kenta S
    Sep 6 at 10:19












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Let, $pi_k(x)$ denote the number of $nle x$ with $k$-prime factors (not necessarily distinct). Using the Sieve of Eratostheness, show that, $$pi_k(x) le fracx(Alog log x+B)^kk! log x.$$ for some positive constants $A$ and $B$.



Im unable to start the problem, as I can not write $pi_k(x)$ is in compact form. If a number $N$ has $k$-prime factors (not necessarily distinct) means if $N=p_1^r_1p_2^r_2cdots p_t^r_t$ then $r_1+r_2+cdots +r_t=k$. So how can I write $pi_k(x)$ in compact form ? Any hint.?










share|cite|improve this question













Let, $pi_k(x)$ denote the number of $nle x$ with $k$-prime factors (not necessarily distinct). Using the Sieve of Eratostheness, show that, $$pi_k(x) le fracx(Alog log x+B)^kk! log x.$$ for some positive constants $A$ and $B$.



Im unable to start the problem, as I can not write $pi_k(x)$ is in compact form. If a number $N$ has $k$-prime factors (not necessarily distinct) means if $N=p_1^r_1p_2^r_2cdots p_t^r_t$ then $r_1+r_2+cdots +r_t=k$. So how can I write $pi_k(x)$ in compact form ? Any hint.?







number-theory analytic-number-theory sieve-theory






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asked Sep 5 at 7:05









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  • maybe $pi(x^1/k)^k$
    – Kenta S
    Sep 6 at 10:19
















  • maybe $pi(x^1/k)^k$
    – Kenta S
    Sep 6 at 10:19















maybe $pi(x^1/k)^k$
– Kenta S
Sep 6 at 10:19




maybe $pi(x^1/k)^k$
– Kenta S
Sep 6 at 10:19















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