## Is this a possibly novel reformulation of the Riemann Hypoythesis?

The Hypothesis described here below is equivalent to the Riemann Hypothesis.

However, I am unable to state whether it is a novel one.
As yet, I do not know whether said equivalent hypothesis could represent an easier challenge than the Riemann Hypothesis, or a more difficult one, or whether it would simply turn the RH into an equally difficult task.

Are there experts on the Riemann Hypothesis who could advise on this? Thanks.

Said possibly novel equivalent reformulation concerns the uniform convergence of a certain ratio of partial sums of the Dirichlet Eta Function, and it can be stated very briefly:

said $\rho=\frac{1}{2}+\alpha+it \; , \; \tau=\frac{1}{2}-\alpha+it$

a pair of arguments falling in the critical strip and symmetrical with respect to the critical line, and

$P_n(\alpha,t)=|S_n(\rho)|/|S_n(\tau)|$

the ratio of the modulus of the n^th partial sums of the corresponding Dirichlet Eta functions, it is shown that proving the sequence of functions $\left\{P_n(\alpha,t)\right\}$ to be uniformly convergent would prove the Riemann Hypothesis, while disproving it would disprove the RH. The hypothesis that the sequence of functions $\left\{P_n(\alpha,t)\right\}$ is uniformly convergent is therefore equivalent to the RH.

For convenience, I just remind that the Dirichlet Eta function is defined as the infinite sum

$
\eta(s) = \sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s} =
1-\frac{1}{2^s}+\frac{1}{3^s}-\frac{1}{4^s}+-\ldots
$

The detailed proof can be found at arxiv.org/abs/0907.2426 (it only requires basic knowledge of undergraduate Calculus)

Luca