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I'm reading Soundararajan's https://arxiv.org/pdf/0705.0723.pdf, and on page 5, one has

$$\sum_{n\leq x} \frac{\Lambda(n)}{n^z} \log (x/n) = -\frac{\zeta'}{\zeta}(z)\log x - \Big(\frac{\zeta'}{\zeta}(z) \Big)' -\sum_{\rho} \frac{x^{\rho-z}}{(\rho-z)^2} + O(1/T),$$ where $\Lambda$ denotes the von Mangoldt function, $\zeta$ the Riemann zeta function, $\zeta(\rho)=0, \Re(z) \in (1/2, 2], x\geq 2$ and $|\Im(\rho)|\leq T$. Maybe i'm missing something, but doesn't the right hand side of the above formula have a pole at $z=1$, which would render the formula meaningless at $z=1$?

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The formula you quote is on page 8 (not page 5). The sum is over all zeros $\rho$, not just those with $|\Im(\rho)|\leq T$. On the other hand, there is the assumption $\Im(z)\in[T,2T]$.

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    $\begingroup$ @user164760: Soundararajan will know when he proves the Riemann Hypothesis. $\endgroup$
    – GH from MO
    Commented Sep 3, 2020 at 22:25
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    $\begingroup$ $F(\sigma+it)$ was defined on page 7. He does not bound $F(\rho)$, because $F(\rho)$ is undefined. You mixup his notation and thoughts with your notation and thoughts. That's rather dangerous. What Sound proves is this: $$\sum_{n\leq x} \frac{\Lambda(n)}{n^{\sigma+it}\log n}\log(x/n) + O(1) = (\log x)\log \zeta(\sigma + it) + \frac{\zeta'}{\zeta}(\sigma+it) - \sum_\rho\dots,$$ where $\sum_\rho\dots$ is an explicit sum over the zeros. And, in the next line, he bounds $|\sum_\rho\dots|$ by a product involving $F(\sigma+it)$. $\endgroup$
    – GH from MO
    Commented Sep 3, 2020 at 22:40
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    $\begingroup$ @user164760: I can assure you that it does not. Please don't try to prove the RH until you have basic questions like the ones above. Start with small things, and grow them big over a long time. $\endgroup$
    – GH from MO
    Commented Sep 3, 2020 at 23:29
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    $\begingroup$ @user164760: You misunderstand the purpose of this site. It is not for discussions, where I say something and then you say something, and we cycle. It is also not for checking proofs, especially of major conjectures. Discuss your proofs with your friends, professors or colleagues, or submit them to journals. That's how these things work. $\endgroup$
    – GH from MO
    Commented Sep 3, 2020 at 23:39
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    $\begingroup$ @user164760: Just to explain perhaps why GH wouldn’t “pinpoint the error”: Asking a more experienced mathematician to find problems in your proofs is like asking a plumber to explain how to fix your plumbing. You’re asking them to spend time doing part of their job, for free — so they’ll often be happy to chat for a bit, especially if it’s an interesting topic (which is why MO works well), but if you come back repeatedly, then they will get tired at some point and want to move on. If you keep asking questions as long as they keep answering, then the only way to move on is to stop answering. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Sep 4, 2020 at 8:01

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