1
$\begingroup$

I know that $\sum_{p \leq N} \frac{1}{p} \geq \log\log N -1 $

However, want to show that $\sum_{p \leq x} \frac{1}{p} \geq \log\log x -1 $.

If let $N=[x]$, then we get a bound for x, i.e. $N \leq x . However, from that all I can seem to get is this $\sum_{p \leq N+1} \frac{1}{p} \geq \log\log(N+1) -1 >\log\log x-1 $

Can't seem to be able to conclude that. As we have $\frac{1}{N+1}$

  • 2
    You are right, the assertion with $x$ is a very tiny bit stronger than the assertion with $N$. It would certainly be possible to find a function $f$ such that $f(N) \ge \log\log N-1$ but $f(N) <\log\log (N+1/2) -1$. One would have to look back on the proof of the result for $N$ and see what kind of slack there is. Since $\log\log$ grows with icy slowness, not much slack is needed.2012-02-09
  • 1
    Your post seems to have died in mid-air. Anyway, maybe you want to look beyond the *fact* $\sum^N\ge\log\log N-1$ and study a *proof* of the fact and see whether the proof can be made to work for $\sum^x$.2012-02-09
  • 0
    I cleaned up the TeX code. Please don't write $log log N$ if you mean $\log\log N$. The code for the former is "log log N"; for the latter it's "\log\log N".2012-02-09

1 Answers 1

1

Well, in your other thread we got $\displaystyle \sum_{p \leq N} \frac{1}{p} \geq \log\log(N+1) -1\ \ $ (with $N \to N+1$)

and this is enough to get $\displaystyle \sum_{p \leq x} \frac{1}{p} \geq \log\log x -1\ \ $ (for $N=\lfloor x \rfloor$)

  • 0
    Thanks for that. I was doing something really stupid. You have been really helpful again.2012-02-09
  • 0
    @simplicity: I have a doubt now, it seems we got rather $\displaystyle \sum_{p \leq N} \frac{1}{p-1} \geq \log\log(N+1)\ $ instead... I'll have to look at this again tomorrow...2012-02-09
  • 0
    @simplicity: I updated my answer in the [other thread](http://math.stackexchange.com/questions/105714/analytical-number-theory/105728#105728) to get your initial inequality.2012-02-09