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comment 
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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When is 2 a generator for a prime modules?
Search for info on "Artin's primitive roots conjecture". 
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. 
Feb 26 
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Averages of $L(s,\chi)$
In what sense is convergence intended? That is how do $m$ and $n$ go to infinity? 
Feb 25 
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Most dense subset of numbers that avoids arbitrarily long arithmetic progressions
If you're interested in lower bounds for the largest set without a $k$AP, then Behrend's construction is still essentially the best known, and it gives a set of size $n\exp(c\sqrt{\log n})$ for some constant $c>0$. See this recent paper of Green and Wolf: arxiv.org/pdf/0810.0732v1.pdf 
Feb 15 
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Does the Euler product converge at $s=1$ for the Dirichlet $L$ function?
All the standard books (e.g. Davenport) will discuss bounds for $\psi(x,\chi)$ for a nonprincipal character $\chi$, from which you can obtain the desired convergence (on the $1$line) by partial summation. Convergence on any other line in the critical strip is unknown, being equivalent to a quasiRiemann hypothesis. 
Feb 8 
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Word complexity of primes mod 4
See this related MO question: mathoverflow.net/questions/168378/… . So far as I know, the only nontrivial result is Shiu's theorem which implies that arbitrarily long strings of $0$'s (or arbitrarily long strings of $1$'s) appear in this word. 
Jan 30 
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Is $x^px+1$ always irreducible in $F_p[x]$?
This is Exercise 13.5.5 in Dummit and Foote. 
Jan 30 
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Primes and Parity
Related MO question (and see rlo's answer there): mathoverflow.net/questions/164936/… 
Jan 20 
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Mean value of Maass forms
@paulgarrett: Sure, that works too! (I wrote the first thing that came to mind.) 
Jan 20 
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Mean value of Maass forms
@paul garrett: Once you approximate the characteristic function by $f$, just use Cauchy's inequality to bound $\langle \phi_j, f\chi_A \rangle$ by $\le \Vert f\chi_A\Vert$. (I edited that in to the answer) 
Jan 19 
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Thin sequences with good counting properties
@juan When the $a_n$ are distinct, then from the conditions it follows that very few of the $a_n$ can be multiples of a prime $p$ with $p\le x^{1/10}$ say. But then the number of integers up to $x$ composed of primes larger than $x^{1/10}$ is $O(x/\log x)$. 
Jan 19 
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Thin sequences with good counting properties
If the $a_n$ can have repeats, then you should just be able to repeat primes an appropriate number of times. If the $a_n$ are distinct, then there is no such sequence. Perhaps you could clarify the question. 
Jan 10 
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Binomial coefficient identity
Look at $\int_0^1 (1x)^m x^{n1} dx$ and use the binomial theorem together with the beta integral. 
Jan 2 
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Are there any serious investigations of whether “mathematicians do their best work when they're young”?
Perhaps this should be communitywiki? It seems opinionbased, and doesn't seem to me to admit a unique, definitive answer. 