By tightening the bounds on $S(t)$ and $S_1(t)$, you could prove conjecture 2 directly. But this bounds are still a mystery...

There is stronger evidence against my conjecture: Karatsuba showed that certain bounds on $|\zeta(s)|$ in infinitessimally short intervals near the crirical line, imply bounds on the multiplicity of zeros. This seems to show that as a function of $t$, the diference $|f(\lambda_n + it)| - |g(\lambda_n + it)|$ would not merge into a constant for infinite t: but it is still far from a refutation, since the multiplicity is merely bounded by a multiple of log $t$, and hence is free to force the difference to vanish, as $t$ grows.

While this is evidence against conjecture 1, it is far from a refutation. We know that $1 -|\tan\theta(t)|$ does not tend to zero on a big chunk of the critical line, where $|\zeta(s)|$ is bounded by the central limit theorem, and yet $1 -|\tan\theta(t)|$ might end up becoming zero as $t$ tends to infinity.

Conrad. Thank you for the bright response. A few comments. It seems to me that you are assuming that $|\zeta(s)|$ is bounded near the critical line, which is a much weaker assumptiom than RH(RH implies a bound, but not viceversa). If, as we come infinitessimally close to the critical line(or to the real part of a zero), $|\zeta(s)|$ becomes unbounded, then it is perfectly possible that the product $|f(t)|(1 -|\tan\theta(t)|$ might tend to zero, as t tends to infinity, even if we assume $\lambda_n$ to be fixed.

Conjecture 1 is that the diference $|f(s)| - |g(s)|$ tends to zero when $s = \lambda_n +it$, where $\rho_n = \lambda_n + i\gamma_n$ runs through all the zero's of the zeta function on the critical strip, and $t$ tends to infinity(not to $\gamma_n$). I do not assume RH. Conrad: are you sure this is false? Has it been proven? My intention is to see if by assuming conjecture 1, the finer shades of the imaginary behaviour of $\zeta(s)$ might be illuminated... Best

When the zeta function is complex, $f$ and $g$ are, respectively, the real and imaginary part of this complex number.

As usual, sigma and t are the real and imaginary parts of the zeta function. I assume that the real part of the zeta function, is restricted to the zeros in and to the right of the critical line, as t tends to infinity. $f(s)$ and $g(s)$ are harmonic functions.

Francois. This is much appreciated! I've been looking in vain for it for some time.

Sorry for the delayed response. I really appreciate the info. I will contact the link and try to get access to the paper. Brun is a Colossus of XXth century mathematics: without him his disciple Selberg would have been quite unknown. It is a shame that his papers are so scattered...