[math-fun] phi(N)*sigma(N)<N^2, phi(N)+sigma(N)>=2*N

The formula for phi(N) is N * product (1 - 1/P) product taken over primes P that divide N. The formula for sigma(N) is N * product (1 + 1/P + 1/P^2 + ... 1/P^E) with P dividing N and P^E being the highest power of P that divides N. So obviously sigma(N) < N * product(1 + 1/P + 1/P^2 + ... ) = N^2 / phi(N). Q.E.D. Also obviously sigma(N) >= N * product ( 1 + 1/P ) so phi(N) + sigma(N) >= 2*N since the cross terms in the products all either cancel out or come out to something positive. [The avoidance of phi+sigma=2*N+5 is not so trivial, though.] -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)