Lagarias’ equivalence to the Riemann hypothesis

[CGUE]

Lagarias’ equivalence to the Riemann hypothesis should be discussed, i.e., if
h_n := n-th harmonic number := 1/1 + 1/2 + · · · + 1/n, and
σ_n := divisor function of n := sum of positive divisors of n, then if n > 1,
h_n + e^h_n ln h_n > σ_n.

There is a $1,000,000 prize for the proof of this at www.claymath.org

 

[mhill]

For me this approach is a bit of nonsense, since you can not evaluate the divisor function for every n=0,1,2,3,4,... not even an asymptotic formula (with a good remainder) is known for divisor function

I think that the most promising approach will come from Hilbert-Polya conjecture or the condition of a Fourier transform having only real zeros.