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1d

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Is the set $ AA+A $ always at least as large as $ A+A $?
The claim for subsets of the integers holds, and follows from this MO question: mathoverflow.net/questions/168844/… 
1d

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Density of polynomials which are soluble with respect to a set of primes
It's still a tiny bit tricky. What you're thinking of is that a random polynomial, which will have Galois group $S_n$ typically will have this feature as you vary over $p$. But for a fixed prime, there is a positive probability that the discriminant is a multiple of $p$, and this I think will affect some calculations. I didn't think this through fully, so maybe you're fully correct, but it's just not immediate why. 
1d

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Density of polynomials which are soluble with respect to a set of primes
Some qualification of this answer seems needed. The question fixes the degree $n$. For example when $n=1$ or $n=2$ this answer is not correct. 
Apr 24 
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Asymptotic limit of truncated Legendre sieve
@user45947: You're right that this is insufficiently well known. For example, I think I should have known this, but I didn't! One lives and learns. 
Apr 23 
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Asymptotic limit of truncated Legendre sieve
@TerryTao: Yes indeed it is $1$. I should have thought an epsilon more! 
Apr 21 
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Is this theorem on $L$functions known?
It seems to me that some result like yours would follow from playing around with the Hadamard factorization theorem. I think you should look carefully at the literature. While it may be hard to find exactly what you've stated, you'll find that many related interesting questions, and maybe you'll have ideas on some of them. Good luck! 
Apr 20 
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Small quotients of smooth numbers
I think this is a very interesting question, and don't believe it is known. Note that it implies your other question on logarithms of ratios of squarefree numbers, which also is unknown I think. Finally, it may be worth pointing out that the kind of bound you are asking for is best possible  random products of the first $k$ primes will cluster, and then use pigeonhole to find two near each other. 
Apr 15 
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When has the BorelCantelli heuristic been wrong?
How about the Maier phenomenon that occasionally there are intervals around $x$ of length $(\log x)^{100}$ say with substantially more (or fewer) primes than one would expect? 
Apr 14 
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distribution of $\sqrt{1} \mod p$
Hooley proved the equidistribution of roots $\mod d$ for composite $d$. That is a much easier problem than the one resolved by Duke, Friedlander and Iwaniec. 
Apr 14 
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distribution of $\sqrt{1} \mod p$
I think they want the determinant to be positive; ie the quadratic polynomial has complex roots as in $x^2+1$. Also I don't see how uniform distribution of the angle helps, since we need the distribution of $ab^{1} \pmod p$. 
Apr 14 
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Zeros of Polynomial with decreasing coefficients
This doesn't seem right. Maybe you're thinking of something else: note the exponents $n_1$, $\ldots$, $n_m$ need not be consecutive. 
Apr 5 
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“frequency” of fields for which the padic regulator vanishes (mod p)
I hope people were not downvoting this simply because of tex. 
Apr 4 
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Historical (personal) examples of teachingbased research
That's very nice  shame it was stupidly removed from wikipedia. 
Mar 26 
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Semiprime number theorem with small prime factor
The typical semiprime will have one very large prime bigger than $N^{1\delta}$ for any $\delta>0$, and one small prime (below $N^{\delta}$). So the same asymptotic holds in your question, with the same proof. 
Mar 19 
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Cricket and the HardyLittlewod maximal function
@BenGreen: I did think that you might like this answer, except for the last line  sorry! 
Mar 9 
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Euler series with milder divergence
How about primes with a prime number of digits? 
Mar 9 
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Factorization when a factor is partially known
Why the votes to close? It seems an interesting problem that people have worked on. 
Mar 7 
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Averages over integer points of the sphere
@Asaf: What you say is not quite correct. There are several similar results that follow from the works of Iwaniec and Duke. Equidistribution of lattice points on the sphere, Heegner points, and equidistribution of closed geodesics. For the equidistribution of closed geodesics analog indeed ELMV give a proof based on Linnik's ideas, but for the problem discussed here GH from MO has described the situation carefully. 
Mar 5 
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Why would the roots of the generating functions of the number of kalmost primes less than x have negative real parts?
@KevinSmith: I added some clarifications to the answer. 
Mar 1 
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Regularized sums of Mobius sequence
@JoeSilverman: I added a clarification above. 