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20h

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Lower bound of first moment of $L$function on $\mathrm{GL}(3)$
Yes, this kind of lower bound in $t$aspect is trivial for any $L$function. The point is that you can approximate $L(\frac 12+it)$ by some long Dirichlet polynomial $\sum_{n\le T^{r}} a(n)n^{1/2it}$, say, with $a(1)=1$, and the $L$function coming from $GL(r)$, say. If you now integrate with smooth weights $L(1/2+it)$ (without absolute values), then note that only the term $n=1$ contributes. The rest of the terms cancel out and are negligible for smooth weights (rapid decay of Fourier transforms). So the bound $\gg T$ follows. 
2d

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Is fourier analysis necessary to prove this?
For part 2 note that the term $x=0$ is $1$, and the rest of the terms are at least $2 \sum_{n=1}^{\infty} e^{\pi n^2} = 0.086\ldots$. 
Oct 15 
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Question about non trivial zeros of Riemann zeta function
Voting to close as equivalent to RH. Use $\sum_n z^n/n=\log(1z)$. 
Oct 10 
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The periodic architecture underlying the natural numbers
I'm voting to close since I don't see a real question here. 
Oct 5 
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Why do roots of polynomials tend to have absolute value close to 1?
The pdf of the Science Direct article is in fact free at the link that you gave. Articles in many Springer & Elsevier journals are freely available four years after publication (and the same holds for all AMS journals). 
Sep 19 
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Number of solutions in a sum of squares Diophantine equation
For fixed $p\ge 3$ the answer is correct and the bound is tight, as you note. But for $p=2$ you need an extra $\log n$ factor. 
Sep 19 
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Inequality due to Siegel (assumptions) and upper bounds on number field discriminants
The inequality in (*) should give a lower bound for $d$ rather than an upper bound. It looks like you're interested in lower bounds for discriminants. Look up Odlyzko's survey article on this (available from his website). 
Sep 19 
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Is there a von Kochtype theorem for the generalized Riemann hypothesis?
@KConrad: Yes the last para is for a fixed $q$. But note that $\psi(x;q,1)$ is being compared to $\psi(x)$ rather than $x$. So in this formulation I can get away without assuming RH. 
Sep 19 
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Is there a von Kochtype theorem for the generalized Riemann hypothesis?
@GHfromMO: Thanks; I didn't know that before! 
Sep 18 
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What are all positive integers n for which the congruence $a^{n+1} \equiv a (mod n)$ holds?
The edited question makes all the earlier comments and answers incomprehensible. This is not a good way to proceed. 
Sep 15 
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$x^2+1$ attaining almost prime values
Most of the numbers with $r$ prime factors have $r1$ small prime factors and one large prime factor. Given that, I think it would be difficult to evaluate $N_r(x)$, since eventually you will be reduced to estimating primes in a quadratic sequence. I don't quite understand your last sentence: any sieve method will give an upper bound of the right order. Maybe you mean right constant as well? Finally there's a difference between counting numbers with $r$ prime factors, and weighting with $\Lambda_r(n)$. My comment applies to the first form (the question as written) and not the second. 
Sep 14 
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Solving a non linear equation
If $K$ is reasonably large then the fixed point seems very close to $K^{2/3}$  even for $K=6$ this is a pretty close approximation. Some asymptotic analysis around here (which might be unpleasant but straightforward) should settle the issue  first show that the fixed points have to be reasonably close to $K^{2/3}$ and then use the derivative to check uniqueness. 
Sep 14 
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On the maximum number of $t$subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections
In your statement of Wilson's theorem don't you need the condition that $t1$ divides $n1$? 
Sep 13 
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On the maximum number of $t$subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections
@DouglasZare: Actually my answer there gave an explicit upper bound which is tight in some cases. 
Sep 13 
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On the maximum number of $t$subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections
See mathoverflow.net/questions/161159/… 
Sep 12 
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Can a sum of roots of unity be an integer?
Certainly missed that! I'm curious how you came across this problem. 
Sep 9 
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The maximum of the preimage of [1,x] through Euler's totient function
Choose the largest primorial below $\sqrt{x}$, and find a multiple of this close to $e^{\gamma} x\log \log x$. That does the job (together with the classical lower bound for $\phi(n)$). 
Sep 9 
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What is the status of the Gauss Circle Problem?
Today's arXiv posting by Shaneson arxiv.org/pdf/1409.2446.pdf indicates that he and Cappell were unable to find an ``errorfree version" of their announced proof. 
Sep 9 
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Can a sum of roots of unity be an integer?
That's very nice! 
Sep 5 
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Can a sum of roots of unity be an integer?
The Evans paper is interesting, but I haven's studied it fully yet. I also found a simple proof for all prime powers, and will write this up maybe later today. 