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 Apr 13 comment Conjugacy classes of $SL_2(Z)$ Extraordinary: the paper which has Chowla as an author, was also communicated by Chowla! Apr 12 comment Asymptotic value of sum over Möbius function You're just asking for square-free numbers below $x$ with no prime factor larger than $\sqrt{x}$. The answer is $\sim (6/\pi^2)x (1-\log 2)$, where $(6/\pi^2)$ comes from the square-free-ness, and $1-\log 2$ from having no prime factor larger than $\sqrt{x}$. Apr 10 comment Reference Request on the existence of $k$ satisfying $P(\Phi_d(2))^k \gt \Phi_d(2)$ for all $d$ I don't think there would be any $k$ for which this holds: one should expect that $\Phi_d(2)$ is pretty smooth every once in a while. What one can prove here is very weak. See my answer to mathoverflow.net/questions/199599/… which links to the paper of Stewart; his Theorem 1 gives the best known lower bounds towards your problem, and it's much weaker than what you want (but as I said what you want is probably false). Apr 6 comment Any heuristics explaining why one seems to have $2n=p+q\Rightarrow\pi(p)+\pi(q)=Li(2n)+o(1)$? Dear @Joël: One can in fact resolve your question in the negative. The point is that there will be many Goldbach solutions with $p$ and $q$ both pretty close to $n$ (at least for many numbers $n$) and then $2 \text{Li}(n)$ (which will be a very good approximation to $\pi(p)+\pi(q)$) does differ significantly from $\text{Li}(2n)$ (in terms of size $n/(\log n)^2$). Apr 1 comment What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky? The Executive Editor for Math Reviews has provided a relevant portion of a review which would otherwise not be freely available to everyone. This seems worthwhile to me, even if it doesn't answer the question fully. Why are people downvoting this answer? Mar 14 comment Could this unexpected bias in the distribution of consecutive primes have any impact on the security of encryption algorithms? Hard to see how. Mar 8 comment What was a cusp to Hurwitz in 1892? This is an interesting question for MO -- I would prefer it stay here than get migrated elsewhere. Feb 28 comment Can the Dedekind zeta function distinguish between real and imaginary quadratic number fields? Not clear what you mean by an algorithm here. What is the input size? For example, you could divide your series by zeta, and get a Dirichlet L-function and then check to see if you can identify the period of those coefficients etc. Sounds like an interesting question, but it might need to be made more precise. Feb 23 comment A converse of the abc conjecture? Well if you assume everything in Mochizuki, Vesselin Dimitrov has an arXiv preprint with strong consequences. Feb 14 comment What is the best way to learn about Modular Forms? Try the second half of Serre's Course in Arithmetic. Feb 7 comment If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal This follows just by partial summation, and would have been known to everyone. For example, how did Chebyshev know that the limit in the prime number theorem, if it exists, must be one? Feb 5 comment Asymptotics of product of Euler's totient function (A001088)? Yes and it equals $\frac{1}{e}\prod_p (1-1/p)^{1/p}$. Feb 2 comment natural radical and an algebraic expression in $\pi$ and/or $e$ I didn't really have anything to add to that one line. By all means include it in the question. (Ok here's one more line to go with my comment: the same constant also appeared in an old asymptotic formula of Bateman to count the number of integers $n$ for which $\phi(n)\le x$. This is why I recognized the Euler product at once, but it is just a coincidence.) Jan 30 comment natural radical and an algebraic expression in $\pi$ and/or $e$ $\zeta(2)\zeta(3)/\zeta(6)$. Jan 27 comment Upper bound on answer for Pell equation You only get from this a bound of $\exp(p^{1/2+\epsilon})$. Please see argument in my answer below. Jan 15 comment Characters of permutation groups @GjergjiZaimi: You're probably right. I just didn't know what to call it! But Polya + cycle index covers the bases. Jan 5 comment Does the antidiagonal in this square matrix always contain a prime? See mathoverflow.net/questions/217956/… which summarizes what is known. Dec 29 comment A combinatorial problem The question at present reads fine to me, but note that this is your fourth attempt at formulating this (in under one hour), and perhaps you could have put in the effort to formulate the question carefully before posting. I can't know of course, but that may have influenced the down vote/close vote. Dec 29 comment Asymptotic growth rate of coefficients of generating function Just because the radius of convergence is $\rho$ does not mean that $S(z)$ goes to infinity as $z\to \rho$. For example, consider $\sum_{n=1}^{\infty} z^n/n^2$. Dec 27 comment Asymptotic growth rate of coefficients of generating function Since only odd powers appear here, you probably also want to add in a contribution of the form $(-\rho)^{-n}$ etc.