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Q: Evaluate a kronecker symbol sum: $\sum\limits_{n=1}^\infty \frac {\big(\frac n x\big)}n$ and $\sum\limits_{n=1}^\infty \frac {\big(\frac xn\big)}n$

Tyma GaidashThe Kronecker Symbol $\left(\frac nm\right)$ has a range of $\{-1,0,1\}$ and $\sum\limits_{n=1}^\infty\frac{(-1)^n}n=-\ln(2)$, so we combine to find the following with the using software: $$\sum_{n=1}^\infty \frac {\left(\frac 2n\right)}n=\sum_{n=1}^\infty \frac {\left(\frac n2\right)}n=\frac{\si...

 
 
15 hours later…
4:55 PM
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Q: Is true that $f:∐_{σ∈Σ}X_σ⟶X$ is continuous iff $f∘ι_σ$ is continuous when $ι_σ$ is the standard embedding of $X_σ$ into $∐_{σ∈Σ}X_σ$?

Antonio Maria Di MauroGiven a collection of topological spaces $$ \mathfrak X:=\{X_\sigma:\sigma\in\Sigma\} $$ we let consider the natural inclusion $$ \iota_\sigma: X_\sigma\ni x_\sigma\longrightarrow(x_\sigma,\sigma)\in\coprod_{\sigma\in\Sigma}X_\sigma $$ Well I would like to understand if a function $f$ form $\copr...

 

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