Mysterious calculator exercise (1 Viewer)

[tywebb]

Try this on your calculator. Nobody knows why it works.

First we need some definitions.

Suppose d_n=divisor function of n=sum of positive divisors of n

Hence

d₁=1

d₂=1+2=3

d₃=1+3=4

d₄=1+2+4=7

d₅=1+5=6

d₆=1+2+3+6=12

etc.

And suppose h_n=n-th harmonic number=1/1+1/2+1/3+...+1/n

So

h₁=1/1=1

h₂=1/1+1/2=3/2 (alternatively h₂=h₁+1/2=3/2)

h₃=1/1+1/2+1/3=11/6 (alternatively h₃=h₂+1/3=11/6)

h₄=1/1+1/2+1/3+1/4=25/12 (alternatively h₄=h₃+1/4=25/12)

h₅=1/1+1/2+1/3+1/4+1/5=137/60 (alternatively h₅=h₄+1/5=137/60)

h₆=1/1+1/2+1/3+1/4+1/5+1/6=49/20 (alternatively h₆=h₅+1/6=49/20)

etc.

And define f_n=n-th Lagarias' number=h_n+e^h_nlnh_n.

Then we have the

Proposition For all n>1, f_n>d_n.

And this is where the calculator comes in.

Check on the calculator for example f₂>d₂:

f₂=3/2+e^3/2ln(3/2)=3.32>d₂=3.

And likewise for others:

f₃=11/6+e^11/6ln(11/6)=5.62>d₃=4

f₄=25/12+e^25/12ln(25/12)=7.98>d₄=7

f₅=137/60+e^137/60ln(137/60)=10.38>d₅=6

f₆=49/20+e^49/20ln(49/20)=12.83>d₆=12

etc.

Nobody knows why this works because the proposition is equivalent to the Riemann Hypothesis!

Lagarias proved equivalence in 2001: http://arxiv.org/PS_cache/math/pdf/0008/0008177v2.pdf