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comment Semiprime number theorem with small prime factor
The typical semi-prime 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
comment When is 2 a generator for a prime modules?
Search for info on "Artin's primitive roots conjecture".
Mar
19
comment Cricket and the Hardy-Littlewod maximal function
@BenGreen: I did think that you might like this answer, except for the last line -- sorry!
Mar
9
comment Euler series with milder divergence
How about primes with a prime number of digits?
Mar
9
comment Factorization when a factor is partially known
Why the votes to close? It seems an interesting problem that people have worked on.
Mar
7
comment 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
comment Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts?
@KevinSmith: I added some clarifications to the answer.
Mar
1
comment Regularized sums of Mobius sequence
@JoeSilverman: I added a clarification above.
Feb
26
comment Averages of $L(s,\chi)$
In what sense is convergence intended? That is how do $m$ and $n$ go to infinity?
Feb
25
comment 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
comment 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 non-principal 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 quasi-Riemann hypothesis.
Feb
8
comment Word complexity of primes mod 4
See this related MO question: mathoverflow.net/questions/168378/… . So far as I know, the only non-trivial 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
comment Is $x^p-x+1$ always irreducible in $F_p[x]$?
This is Exercise 13.5.5 in Dummit and Foote.
Jan
30
comment Primes and Parity
Related MO question (and see rlo's answer there): mathoverflow.net/questions/164936/…
Jan
20
comment Mean value of Maass forms
@paulgarrett: Sure, that works too! (I wrote the first thing that came to mind.)
Jan
20
comment 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
comment 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
comment 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
comment Binomial coefficient identity
Look at $\int_0^1 (1-x)^m x^{n-1} dx$ and use the binomial theorem together with the beta integral.
Jan
2
comment Are there any serious investigations of whether “mathematicians do their best work when they're young”?
Perhaps this should be community-wiki? It seems opinion-based, and doesn't seem to me to admit a unique, definitive answer.