Hi, I would be grateful for any comments pointing out gross mistakes in this "proof" as it seems too simple (and short!) to be correct. Did I miss something? Wrong assumptions? Is it common knowledge, and fails for large values of b? I tried to double check each step numerically and it seems correct within machine precision. Thank you.
The reflection functional equation for the Riemann zeta function is
where and . For simplicity, restrict what follows to , and multiply each side of (1) by its complex conjugate to obtain:
Using and the Ramanujan identity:
where the infinite product in (3) converges absolutely for , substitute in (2) to obtain:
The left-hand side of (4) is identically equal to 1 for all values of b when a= 1/2 (i.e. on the critical line) and for all other values of a, b on the critical strip 0<a<1 that are zeroes of since the zeroes are symmetrically distributed about the critical line.
Take the logarithm on each side of (4) and its partial derivative with respect to a,
Inside the critical strip 0<a<1, b>0, the first three terms on the right-hand side of (5) are positive, the 4th term is bounded between approximately -1.2 and 0.9, and the infinite sum term is always negative.
For small values of b (between 0 and slightly less than 2 pi (by my estimate)), the positive terms on the RHS of (5) dominate for all 0<a<1 and the derivative is uniformly >0 hence only on the critical line. For larger values of b (slightly more than 2 pi), the negative infinite sum dominates the RHS of equation (5) for all b, and also only on the critical line.
Hence the above would seem to show that there are no zeroes off the critical line except for a small region near b= 2pi which I did not scrutinize here since others have already shown the RH to be true up to a much greater ` lower bound` for b.