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If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{n^s}$$ sometimes works on the abscissa of convergence of the ordinary Dirichlet series.

Let $\rho=\beta+i\gamma$ denote non-trivial zeros of $\zeta(s)$. By truncating the integral representation and estimating the tail, one can prove that for $\sigma\geq \sup \beta $ we have $$\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{\Lambda(n)-1}{n^s}=-\frac{\zeta'(s)}{\zeta(s)}-\zeta(s)+O\left(\frac{1}{\log x}\right)$$ except when $s=\rho$, at which points the sum diverges like $\log x$.

I have seen some graphs of the partial sums when $s=1/2+it$ compared with $$-\frac{\zeta'(1/2+it)}{\zeta(1/2+it)}-\zeta(1/2+it)$$ and the results are quite compelling. However, if the Riemann hypothesis is false, divergent terms of modulus $$\frac{x^{\beta-1/2}}{\log x |\gamma-t|}$$ must be added for every zero with $\beta>1/2$, and these will only show up on a graph with sufficiently many terms, or for sufficiently large $t$. Worse still, if there is a conspiracy between the imaginary ordinates of zeros not on the line, these terms may not show up until later.

My question is as follows:

Has anybody actually studied the resemblance the Cesaro summation has to the function at the coarsest scale we can expect divergent terms to be visible (that is, in the absence of a conspiracy, and based on the fact that the first $10^{12}$ or so zeros are known to be on the line)? If so, who did so, and who must I ask if I want to see the results?

I must point out that I am not suggesting these observations actually have any bearing on our understanding of the truth of the hypothesis. I am just wondering if anyone has investigated the behaviour of the Cesaro sum to a degree of accuracy that could possibly reveal divergence in the simplest imaginable case.

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