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13h

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Solution or Reference Request for a Closed Form of the Sum
Do not expect any simple expression for the Legendre function. It is a complicated function in the sense that we cannot answer (even approximately) simple questions like: for a given $q$, how small is the smallest $a\geq 1$ with $\left(\frac{a}{q}\right)=1$ (cf. en.wikipedia.org/wiki/…). We like quadratic reciprocity (among other things) because it allows us to calculate the Legendre symbol in polynomial time. 
13h

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Solution or Reference Request for a Closed Form of the Sum
@user170039: See my comment to your original post. 
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answered  Solution or Reference Request for a Closed Form of the Sum 
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Lfunction of twist
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Expression and growth bound for $r_{p^m,k}(n)$
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answered  Expression and growth bound for $r_{p^m,k}(n)$ 
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answered  Inequality due to Siegel (assumptions) and upper bounds on number field discriminants 
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Inequality due to Siegel (assumptions) and upper bounds on number field discriminants
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Inequality due to Siegel (assumptions) and upper bounds on number field discriminants
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Holomorphic HoffsteinLockhart
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Holomorphic HoffsteinLockhart
@RicardoMenares: Small correction to my previous comment. The factor is not exponentially decaying but exponentially increasing in the Maass case. 
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Holomorphic HoffsteinLockhart
@RicardoMenares: Note also that in the Maass case, we have the same volume factor, and also a factor that decays exponentially in the spectral parameter and takes the role of the factors depending on $k$ above (HoffsteinLockhart build this factor into their Whittaker function, see their (0.1) and the display below it). So the Maass case is not that much different from the holomorphic case. 
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Holomorphic HoffsteinLockhart
@RicardoMenares: Indeed, I forgot about the volume factor. I fixed my last 3 displays accordingly. Thanks! 
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Holomorphic HoffsteinLockhart
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Is there a von Kochtype theorem for the generalized Riemann hypothesis?
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Davenport's proof that almost all integers are the sum of 4 cubes
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Probability question (win three games in a row = win or 4 wins total = win)
This site is for research level questions. For general questions in mathematics see math.stackexchange.com 
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Is there a von Kochtype theorem for the generalized Riemann hypothesis?
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Is there a von Kochtype theorem for the generalized Riemann hypothesis?
Nice observation! 
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Is there a von Kochtype theorem for the generalized Riemann hypothesis?
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