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Hint: Zeta is involved!

\[\large \displaystyle\sum _{ n=1 }^{ \infty }{ \sum _{ k=1 }^{ \infty }{ \dfrac { 1 }{ \left\lfloor \sqrt { n+k } \right\rfloor ^{ a } } } } \]

Find a closed form for the above summation for \(a>4\).


This is a part of the set Formidable Series and Integrals

Note by Hummus A
5 months, 2 weeks ago

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we have \[\displaystyle\sum _{ n=1 }^{ \infty }{ \sum _{ k=1 }^{ \infty }{ \dfrac { 1 }{ \left\lfloor \sqrt { n+k } \right\rfloor ^{ a } } } } =\displaystyle\sum _{ i=1 }^{ \infty }{ \left( \displaystyle\sum _{ { i }^{ 2 }\le k+n<(i+1)^{ 2 } }{ \frac { 1 }{ \left\lfloor \sqrt { n+k } \right\rfloor ^{ a } } } \right) } =\displaystyle\sum _{ i=1 }^{ \infty }{ \frac { 1 }{ { i }^{ a } } \left( \sum _{ { i }^{ 2 }\le k+n<(i+1)^{ 2 } }{ 1 } \right) } \]

evaluating the inner sum we get \(2i^3+3i^2-i-1\)

so it follows that \[\displaystyle\sum _{ n=1 }^{ \infty }{ \sum _{ k=1 }^{ \infty }{ \dfrac { 1 }{ \left\lfloor \sqrt { n+k } \right\rfloor ^{ a } } } } =2\zeta(a-3)+3\zeta(a-2)-\zeta(a-1)-\zeta(a)\] Hummus A · 5 months, 2 weeks ago

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@Hummus A coool ... :) Aman Rajput · 5 months ago

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May be \[\large 2a\zeta(a-1)\] Aman Rajput · 5 months, 2 weeks ago

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@Aman Rajput i'm getting

\(2\zeta(a-3)+3\zeta(a-2)-\zeta(a-1)-\zeta(a)\)

do you want me to post how i got the answer? Hummus A · 5 months, 2 weeks ago

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@Hummus A Post if you can :) Aman Rajput · 5 months, 2 weeks ago

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@Pi Han Goh No the n&k values are used to clarify the most simple values for a, an infinite series has up to the given below number, not every number possibly out there. I can argue this and be right. Ark3 Graptor · 5 months, 2 weeks ago

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@Ark3 Graptor You misunderstood this question. Pi Han Goh · 5 months, 2 weeks ago

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@Ark3 Graptor Please elaborate :) Aman Rajput · 5 months, 2 weeks ago

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