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bio website stanford.edu/~rjlo
location Palo Alto, CA
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visits member for 4 years, 9 months
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May
15
answered S integral points of an elliptic curve, with S of positive density
May
4
answered A square-squareroot integer race sequence involving primes
May
3
comment A square-squareroot integer race sequence involving primes
Legendre conjectured that there is always a prime between $n^2$ and $(n+1)^2$, and this is generally believed to be true. Assuming this, for any $n$, $g(f(n))=n$, so you're just looking at a random walk.
May
3
comment A question on primitive roots
There's surely a dependence on the choice of the primitive roots (assuming my interpretation of the question is correct). Look at the primes $q$ for which $x_p \equiv a\pmod{n}$, and then choose a different primitive root for each such $q$. Why should the congruence $x_p\equiv a\pmod{n}$ be preserved for almost all of these $q$? It might also be possible to hack together a sequence of $x_p's$ which are not equidistributed modulo any $n$, but I'm too tired to think through the details of this right now.
May
3
comment A question on primitive roots
Seva: Despite the notation, it seems that $q$ varying is more interesting. If we fix $q$ and vary $p$, Dirichlet's theorem shows that each possible value of $x_p$ occurs equally often.
May
2
comment Is the alternating sum of primes $2 - 3 + 5 + . . P_N$ asymptotic to $- N ln N/2$ for even $N$?
And, I guess, regarding Ruiz's purported proof: I haven't read it, but the discussion suggests that it's replacing $p_n$ by the asymptotic $p_n\sim n\log n$. This cannot work: there are many sequences which satisfy all the properties we know primes to have (e.g., the prime number theorem), and yet are such that, using the notation from my answer, $S(N;a,q)/p_N$ can tend to any value in $[0,1]$, or for which the limit doesn't exist, etc. To prove that $S(N;1,2) \sim p_N/2$, you have to rule out the possibility that all small prime gaps occur with odd index. But I have no idea how to do that!
May
2
comment Is the alternating sum of primes $2 - 3 + 5 + . . P_N$ asymptotic to $- N ln N/2$ for even $N$?
Regarding the generalization with $k\geq 2$: Using essentially the same techniques and that $p_{n+1}^{k+1}-p_n^{k+1}=(p_{n+1}-p_n)(p_{n+1}^k+\dots+p_n^k)=(p_{n+1}-p_n)((k+1‌​)p_n^k + O_k(p_n^{k-1}))$, it would again be possible to prove a lower bound of the right order of magnitude for the alternating sum of $k$-th powers of primes.
May
2
awarded  Nice Answer
May
1
answered Is the alternating sum of primes $2 - 3 + 5 + . . P_N$ asymptotic to $- N ln N/2$ for even $N$?
May
1
comment An introduction to sieve method and their application, Cojocaru & Murty
Actually, wait. Unless I'm missing something, their claimed error term is wrong, and it should be $O(1/\log x)$ rather than $O(1/x^{1/4})$ (this is fine for their purposes, as the error on the next line is $O(1/\log x)$ anyway). To see it has to be at least this big, just look at the integral of the error from 3 to 4. To see that it's no bigger, split the integral into two parts, integrating from 3 to $x^{1/2}$, say, and from $x^{1/2}$ to $x$. The latter decays like $x^{-1/8}$, while the former can be bounded by $O(1/\log x)$.
May
1
comment An introduction to sieve method and their application, Cojocaru & Murty
Plugging in the lemma, you'd get an $A_1 \log t$ term. This is the same as $-A_1(1+\log(x/t))+A_1\log x + A_1$. Is that your question?
Mar
18
awarded  Yearling
Mar
17
comment Question about a certain class of primes
Greg: Good point. I was pretty sure there was an easy check, but I didn't bother to think about it.
Mar
17
revised Question about a certain class of primes
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Mar
17
revised Question about a certain class of primes
added 851 characters in body
Mar
17
answered Question about a certain class of primes
Mar
7
awarded  Enlightened
Mar
7
awarded  Nice Answer
Mar
7
comment The behavior of a certain greedy algorithm for Erdős Discrepancy Problem
Fixed! Sorry about that. Apparently all the files I'd hosted at Emory before moving have gone away.
Mar
7
revised The behavior of a certain greedy algorithm for Erdős Discrepancy Problem
added 11 characters in body