[dbaupp]

I think you've flipped the condition: the RH says the Riemann zeta function _only_ has zeros along the line 1/2 + iy. (And, indeed, there are known zeros along this line: 1/2 + 14.135... i.)

The Lindelöf hypothesis is, apparently, equivalent to: the number of zeros with real part greater than 1/2+epsilon and imaginary part between y and y+1 is o(log(y)), for any epsilon > 0. That is, boxes of height 1 starting just off the critical line contain few zeros; the RH implies they contain zero.