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Mar
26
reviewed Leave Open How good is “almost all” when it comes to the Riemann Hypothesis?
Mar
26
awarded  Enlightened
Mar
26
awarded  Nice Answer
Mar
26
revised Measure of a set of irrational numbers
added 411 characters in body
Mar
26
revised Measure of a set of irrational numbers
added 411 characters in body
Mar
26
comment Measure of a set of irrational numbers
From properties of continued fractions one knows that $|x-p_n/q_n|$ is roughly $1/(q_nq_{n+1})$ and also $q_{n+1}$ is roughly $a_{n+1}q_n$. So the LHS of the last inequality is about $C/(q_n^2 a_{n+1})$ and now you get that $a_{n+1} \ge c n$. Does that make sense?
Mar
26
answered Measure of a set of irrational numbers
Mar
25
comment Number of representations of an integer as an (arbitrary) sum of products
I think my answer below provides a complete solution to your problem. Were you looking for something more? Or perhaps there was something unclear with what I wrote?
Mar
25
reviewed Close Curves of high genus with many rational points
Mar
24
reviewed Leave Open Consecutive non-quadratic residues
Mar
24
awarded  Enlightened
Mar
23
revised maximum size of intersecting set families
added 761 characters in body
Mar
23
comment Ramanujan's tau function
Also Ramanujan was interested in representations of numbers by quadratic forms, and in particular the number of representations of an integer as the sum of an even number of squares. Exact formulas were known in a number of cases already, and he might have been trying to extend these. As I recall, his famous paper begins by writing down identities for the products of Eisenstein series, and records the first case where one doesn't have an exact identity, and this comes from the tau function.
Mar
23
answered maximum size of intersecting set families
Mar
23
comment maximum size of intersecting set families
Every $k$ element set has $\binom{k}{2}$ subsets of size $2$, and all the subsets of size $2$ from the $m$-sets we have should be distinct. Therefore $m\binom{k}{2} \le \binom{n}{2}$, or $m\le n(n-1)/(k(k-1))$. Are you looking for such upper bounds for $m$, or for constructions giving lower bounds for $m$?
Mar
23
reviewed Close Intuition about poisson summation formula?
Mar
22
reviewed Leave Open Does anyone recognize this generating function
Mar
20
comment Asymptotics of special square-free numbers
See Exercise 4 of Section 7.4 of Montgomery and Vaughan's book on Multiplicative number theory. If $k$ is fixed, then the asymptotics are essentially the same as that of just numbers with $k$-prime factors (forgetting squarefree).
Mar
20
reviewed Leave Open Discriminant of a compositum of number fields, a bound?
Mar
19
answered Angular equidistribution of lattice points on circles