Ivan Shishkin, Rye (1878)

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Divergence de la somme des 1p\frac{1}{p}

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Solution

Solution by mathman · FR

pn1p>pnln(1+1p)=pn(ln(11p2)ln(11p))>ln(ζ(2))+ln(k=1n1k)>ln(ζ(2))+ln(ln(n))\sum_{p\leq n}\frac{1}{p}> \sum_{p\leq n}ln(1+\frac{1}{p})=\sum_{p\leq n}(ln(1-\frac{1}{p²})-ln(1-\frac{1}{p}))> -ln(\zeta(2))+ ln(\sum_{k=1}^{n}\frac{1}{k})>-ln(\zeta(2))+ ln(ln(n))

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