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Let $S=\sum_{n=1}^\infty a_n$, be a semi-convergent series with $T=\sum_{n=1}^\infty a_n^2 < \infty$ and $\sum_{n=1}^\infty |a_n|=\infty$. Under which conditions are the following formulas valid? They are of course valid for finite sums, but not necessarily for infinite sums.

$$S^2 = T + 2 \sum_{n=1}^\infty a_n w_n \mbox{ with } w_n=\sum_{m=n+1}^\infty a_m$$

$$S^2 = T + 2 \sum_{{\substack{\gcd(n,m)=1\\ 0<n<m}}}^\infty v_{n/m} \mbox{ with } v_{n/m}=\sum_{k=1}^\infty a_{km}a_{kn}$$

Things would be straightforward if we were dealing with a Cauchy product, but this is not the case here. Note that in the second formula, the first few terms are $v_{1/2},v_{1/3},v_{2/3}, v_{1/4},v_{3/4},v_{1/5},v_{2/5},v_{3/5},v_{4/5},v_{1/6},v_{5/6},v_{1/7},\dots$.

The result (first formula) is correct for instance if $a_n=(-1)^{n+1} n^{-0.8}$. The series I am interested in are related to the Riemann Hypothesis and defined in the Appendix in my previous question, here.

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    $\begingroup$ "semi-convergent" ... is that also called "conditionally convergent"? If not, what is it? $\endgroup$
    – Gerald Edgar
    Commented Dec 29, 2020 at 12:49
  • $\begingroup$ From what I read on Wikipedia, yes they are the same. More precisely, by semi-convergent, I mean $|\sum a_n|<\infty$ and $\sum |a_n|=\infty$. $\endgroup$
    – Vincent Granville
    Commented Dec 29, 2020 at 18:05
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    $\begingroup$ Possibly of relevance are some papers by Florian Cajori (who did some work on series convergence before devoting himself almost entirely to the history of mathematics, for which he is much better known), such as On the multiplication and involution of semi-convergent series. Other such papers can be found by searching here with Author = Cajori and Title = series (also Title = semi-convergent, without any author being selected). $\endgroup$
    – Dave L Renfro
    Commented Dec 29, 2020 at 18:58

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For $$S^2 = T + 2 \sum_{n=1}^\infty a_n w_n \mbox{ with } w_n=\sum_{m=n+1}^\infty a_m\tag{1}$$ to be valid, it is enough that $\sum_n a_n$ be any alternating series with $\sum_{n=1}^\infty a_n^2<\infty$ and $|a_n|$ converging to $0$ (eventually) monotonically.

Indeed, since (1) holds for finite sums, this question boils down to the convergence of the series $\sum_{n=1}^\infty a_n w_n$. Given the mentioned conditions on $(a_n)$, this series will indeed converge because eventually (for some natural $N$ and all $n\ge N$) we will have $|w_n|\le|a_{n+1}|\le|a_n|$ and hence $\sum_{n=N}^\infty |a_n w_n|\le\sum_{n=N}^\infty a_n^2<\infty$.

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    $\begingroup$ I like this reasoning. Next: what about other conditionally convergent series? $\endgroup$
    – Gerald Edgar
    Commented Dec 29, 2020 at 13:55
  • $\begingroup$ @Iosif: Thank you. Not sure if my series can be called alternating. The signs of the successive terms is not +, -, +, -, +, - etc. but like + - - + + + - + - - - - + + and appear like random. A typical case is $a_n=(-1)^{n+1} \cos(t \log n)/n^{3/4}$ where $t$ is a large real number. That specific example also works. $\endgroup$
    – Vincent Granville
    Commented Dec 29, 2020 at 18:11
  • $\begingroup$ The special series I mentioned is the real part of $(1-2^{1-s})^{-1}\zeta(s)$ in the critical strip $0<\Re(s)<1$, with $\Re(s)=3/4$ and $\Im(s)=t$. $\endgroup$
    – Vincent Granville
    Commented Dec 29, 2020 at 18:38

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