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I was leafing through Gradshteyn–Ryzhik in bed yesterday, as one does, and noticed on the last page that the Mellin transforms of several hyperbolic functions have a factor of $\zeta(s-1)$ or $\zeta(s)$ in them. (That turns out to be very easy to prove — for the functions in that table and several other ones.) That has the following immediate consequence.

If you work out $$\sum \left(1- \tanh\left(\frac{n}{x}\right)\right) \mu(n)$$ (say) or $$\sum \left(1- \tanh\left(\frac{n}{x}\right) \right) \Lambda(n),$$ you will find out that everything turns out beautifully: the zeros of $\zeta(s)$ have no effect, and you get an unconditional expression as a nice, closed-form power series on $x^{-1}$.

I am told (by a non-analytic-number-theorist) that this is known, and a colleague (who is an analytic number theorist) points out that there is a variant that is trivial: $$ \sum_n \frac{\mu(n)}{1-e^{n/x}} = \sum_d \sum_n e^{dn/x} \mu(n) = 1.$$

Still, I'm a bit surprised. Are there any applications?

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    $\begingroup$ I think MO titles should generally avoid clickbait-y tricks ("you won't believe the behaviour of these weighted sums!"), but I quite like this one. $\endgroup$
    – LSpice
    Commented Nov 6, 2022 at 17:22
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    $\begingroup$ The identity $\sum_n \lfloor \frac{x}{n} \rfloor \Lambda(n) = \log(x!)$ (for $x$ an integer) quickly implies Mertens' theorem $\sum_{n \leq x} \frac{\Lambda(n)}{n} = \log x + O(1)$; similarly $\sum_n \lfloor \frac{x}{n} \rfloor \mu(n) = 1$ implies $|\sum_{n \leq x} \frac{\mu(n)}{n}| \leq 1$. That's about the level of what one can accomplish with these sorts of convolution identities between $\mu$ or $\Lambda$ on one hand, and a weight which implicitly involves Dirichlet convolution with the indicator function of the natural numbers on the other (to kill all the contributions of zeta zeroes). $\endgroup$
    – Terry Tao
    Commented Nov 6, 2022 at 17:39
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    $\begingroup$ Given that you are explicitly cancelling out the contribution of the zeroes, I would say that one should only expect to prove results that are insensitive to the location of the zeroes; for instance one would not expect to prove the prime number theorem this way (without some deep Tauberian theorem or something which amounts to establishing non-vanishing of $\zeta(1+it)$). But statements that relate only to the behavior of $\zeta$ near $s=1$ should be attainable from such identities (Mertens' theorem, for instance, is of this form). $\endgroup$
    – Terry Tao
    Commented Nov 6, 2022 at 18:04
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    $\begingroup$ My guess though that your hyperbolic tangent type weights are so smooth that it would be difficult to extract much information from them directly (the identities you mention are likely just smoothly averaged versions of the identities I mentioned previously). On the Fourier side this corresponds to the issue with the decay of the Gamma function that you mentioned. In general it is the rough averages which are the powerful ones - compare the prime number theorem $\sum_{n \leq x} \Lambda(n) = x+o(x)$ with Mertens' theorem $\sum_{n \leq x} \Lambda(n)/n = \log x + O(1)$, for instance. $\endgroup$
    – Terry Tao
    Commented Nov 6, 2022 at 18:26
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    $\begingroup$ One can get elementary proofs of $|\sum_{n \leq x} \Lambda(n) - x| \leq \varepsilon x$ for any $\varepsilon>0$ and sufficiently large $x$ by convolving the identity $\sum_{n} \lfloor \frac{x}{n} \rfloor \Lambda(n) = \log \lfloor x\rfloor!$ with a carefully chosen finite weight depending on $\varepsilon$ in the spirit of Chebyshev, see e.g., Section 9 of projecteuclid.org/journals/… . I would expect the story is similar for Mobius. $\endgroup$
    – Terry Tao
    Commented Nov 6, 2022 at 18:59

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