Timeline for A problem of Erdős on convergence of $\sum (-1)^nn/p_n$ and equidistribution of $\pi(n)$ modulo 2
Current License: CC BY-SA 4.0
12 events
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| Sep 5, 2023 at 17:10 | comment | added | Terry Tao | I was able to use the equivalence mentioned by the OP in a recent paper on this topic. arxiv.org/abs/2308.07205 | |
| Mar 3, 2022 at 10:34 | comment | added | YCor | @User Start from $s_0(n)=n$ (modulo 2). This one is equidistributed and the series converges. Now consider $X=\{2\lfloor n\log n\rfloor:n\ge 4\}$ and $Y=X\cup (X+1)$. Then $Y$ has density zero but $(1/n\log n)$ is already non-summable on $Y$, while the restriction of $(-1)^n/(n\log n)$ converges for partials sums restricted to $Y$. So, define $s(n)=s_0(n)$ for $n\notin Y$ and $s(n)=0$ for $n\in Y$. Since $s=s_0$ on a set of density 1, the equidistribution still holds. But the contribution on $Y$ (and convergence outside $Y$) will force divergence of the series $(-1)^{s(n)}/n\log n$. | |
| Oct 29, 2018 at 11:55 | comment | added | Sean Eberhard | @SylvainJULIEN $|E_n| - |O_n|$ is certainly not bounded, because there are arbitrarily large gaps in the primes. | |
| Oct 29, 2018 at 9:43 | comment | added | Sylvain JULIEN | Intuitively, the convergence requires the sequence $ (-1)^{\pi(n)} $ to be "close to" $ (-1)^{n} $. So perhaps one should find a tight upper bound (probably an $ O(1) $ ) for the quantity $ \vert\vert E(n)\vert-\vert O(n)\vert\vert $. | |
| Oct 28, 2018 at 17:32 | history | edited | Mustafa Said | CC BY-SA 4.0 |
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| Oct 28, 2018 at 16:24 | comment | added | YCor | The convergence cannot be formally deduced if the $1/2$ distribution result holds. To be more precise, it is an exercise that there exists a $\{0,1\}$-equidistributed (in the given sense) sequence $(s(n))$ such that $\sum \frac{(-1)^{s(n)}}{n\log n}$ does not converge. | |
| Oct 28, 2018 at 16:20 | comment | added | YCor | @GerhardPaseman, Mustafa: I suggest to erase all previous comments, obsolete now | |
| Oct 28, 2018 at 16:20 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
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| Oct 28, 2018 at 16:17 | history | edited | YCor | CC BY-SA 4.0 |
added tags, emphasized main series, added accents to Erdős, specified title, emphasized question (which is not Erdos' problem)
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| Oct 28, 2018 at 16:07 | history | edited | Mustafa Said | CC BY-SA 4.0 |
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| Oct 28, 2018 at 16:04 | history | edited | Mustafa Said | CC BY-SA 4.0 |
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| Oct 28, 2018 at 15:57 | history | asked | Mustafa Said | CC BY-SA 4.0 |