((log1)/( (√1)))−((log3)/( (√3)))+((log5)/( (√5)))−((log7)/( (√7)))+..C=((π/4)−(γ/2)−(1/2)log(2π))((1/( (√1)))−(1/( (√3)))+..C) γ =Eulerian constant


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Question Number 122303 by Dwaipayan Shikari last updated on 15/Nov/20

$\frac{l o g 1}{\sqrt{1}} - \frac{l o g 3}{\sqrt{3}} + \frac{l o g 5}{\sqrt{5}} - \frac{l o g 7}{\sqrt{7}} + . . C = (\frac{π}{4} - \frac{γ}{2} - \frac{1}{2} l o g (2 π)) (\frac{1}{\sqrt{1}} - \frac{1}{\sqrt{3}} + . . C)$ $γ = E u l e r i a n c o n s t a n t$
Commented by rs4089 last updated on 15/Nov/20

$i t h i n k i t s e x p a n s i o n i s g i v e n b y i n d i a n m a t h e m a t i c i a n s r i n i v a s a r a m a n u j a n t o p r o f e c e r g . h . h a r d y .$
Commented by Dwaipayan Shikari last updated on 15/Nov/20

$Y e s! B u t i h a v e n o d e r i v a t i o n y e t$
Commented by Dwaipayan Shikari last updated on 15/Nov/20

$A n y I d e a s i r s ? :)$
	Answered by mindispower last updated on 15/Nov/20

	$ζ^{'} (s)$ $= Σ \frac{- l n (n)}{n^{s}}, s e e e m s s = \frac{1}{2}$ $w o r c k i n g o n i t$
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