Kind of weird question, but is there something like an integral operator which returns $1$ if $\gcd(a, b) = 1$ and $0$ otherwise, meaning $$ \int_{D} K(a, b, t) \, {\rm d}t = \begin{cases} 1 \qquad \gcd(a, b) = 1 \\ 0 \qquad {\rm else} \end{cases} \, . $$ where $D$ is some appropriate integration domain and $K$ the kernel. Hope it's clear.
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You can see some solutions here – K.defaoite Jul 02 '24 at 17:32
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Sounds like you've forgotten about the Kronecker delta. I love Lethe, mwahahahahha! Then $$\delta_1^{\gcd(a, b)} = 1$$ if $a$ and $b$ are coprime, and $$\delta_1^{\gcd(a, b)} = 0$$ otherwise.
The Short One
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well... were is the integral ;) PS: let me guess...now you tell me $$\int_0^1 e^{i2\pi t (\gcd(a,b)-1)} , {\rm d} t , ?$$ – Diger Sep 18 '18 at 21:15
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now I tell you to play around with Mathematica or Maple or something like that – The Short One Sep 18 '18 at 21:22
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To see what different (pun intended, shoot me) integrals you can come up with using the Kronecker delta. – The Short One Sep 18 '18 at 21:30
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Hey, this is somewhat 2 years old, but I just wanted to say that with integral operator I meant an operator that does not depend on using the gcd function. – Diger May 24 '20 at 10:05