i707107
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Registered User
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Jun 5 |
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Minimal representation of a polynomial as a linear combination of squares Thanks, I got it now. It can also be done by induction on degree. |
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Jun 5 |
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Minimal representation of a polynomial as a linear combination of squares Why does $h$ have degree at most $n-1$? I know that it should be $\leq 2n-1$, but how do you get $n-1$? |
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Jun 2 |
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Minimal representation of a polynomial as a linear combination of squares When it is reducible with $f=uv$, and degree of $u$, $v$ are $n$, then $p=((u+v)/2)^2-((u-v)/2)^2$. But, not sure how to go for other cases. |
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Jun 2 |
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Sums of inverse determinants over matrices I think the true bound is $S(r)=O(r^2\log r + r^3)$ now. |
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Jun 2 |
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Sums of inverse determinants over matrices I realized that that estimate was overestimate. This is a counting argument for lattice points, like one uses for Gauss Circle Problem. The number of lattice points inside a convex region is approximated by the area of the region + error term depending on perimeter. |
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Jun 1 |
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Sums of inverse determinants over matrices Argument seems to work in general, $S(r)=O(r^{n^2-1}\log r)$. |
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Jun 1 |
answered | Sums of inverse determinants over matrices |
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Jun 1 |
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How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the $S_2$ sum Yes, your outline precisely is C-S. But mine is truncation of sums, so I would have $\sum x_iy_i\leq K\sum x_i+K^{-1}\sum x_iy_i^2$, since the first sum on the RHS is upper bound for the sum of terms with $y_i\leq K$, and the second sum on the RHS is for $y_i>K$. |
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May 31 |
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How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the $S_2$ sum @GH: Good point! I tried reproducing proof of C-S regarding your comment, but misses a factor of 2. So, this method overestimates than C-S by a factor of 2. |
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May 31 |
answered | How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the $S_2$ sum |
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May 29 |
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A technical question related to Zhang’s result of bounded prime gaps @ericc: Edited. Implemented your correction, and simplified a little. I did not need to have $d=d_0 k$, since what I needed is $d$ having many prime factors. |
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May 29 |
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A technical question related to Zhang’s result of bounded prime gaps simplified argument, implemented comment by ericc |
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May 29 |
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A technical question related to Zhang’s result of bounded prime gaps @ericc: Yeah, I missed that. The extra $\tau_{k_0}(d)$ will give extra powers of $\mathcal{L}$ on the numerator. But, my argument still works as you noticed. |
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May 29 |
accepted | A technical question related to Zhang’s result of bounded prime gaps |
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May 28 |
answered | A technical question related to Zhang’s result of bounded prime gaps |
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May 25 |
accepted | Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$? |
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May 24 |
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Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$? Hold for series convergent with radius 1. Any series obtained from your setting, will have radius of convergence 1, or infinity. The case when radius of convergence infinity, is only when $r$ has finite decimal expansion. |
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May 24 |
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Full $n$-torsion of elliptic curves and the cyclotomic field of order $n$ Yes. Weil pairing applies to $n$-torsion too. |
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May 24 |
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Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$? As I mentioned in my answer, $r$ can be either rational or irrational. When $r$ is rational, the function $f$ is rational, and when $r$ is irrational, the function $f$ is transcendental. |
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May 24 |
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Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$? This might be better expression: $$\sum_{p\textrm{ prime}} x^p$$ |
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May 24 |
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Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$? added 74 characters in body |
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May 24 |
answered | Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$? |
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May 16 |
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Ramification in Division field of Abelian Varieties Maybe I should have put, $m$ avoids all primes of bad reduction. |
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May 16 |
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Ramification in Division field of Abelian Varieties Thank you. This is also very clear. So $\mathbb{Q}(E[2])=\mathbb{Q}(\sqrt{d})$, and primes ramify in $\mathbb{Q}$ are precisely the prime divisors of $d$. |
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May 16 |
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Ramification in Division field of Abelian Varieties Thank you for clear counterexample. I'd like to know what happens when $m$ is a power of $p$. |
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May 16 |
asked | Ramification in Division field of Abelian Varieties |
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May 15 |
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Sums of uniformly random vectors from the $n$-dimensional unit ball To clarify: $||v_i||\leq 1$ right? not $||v_i||=1$? |
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May 9 |
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Least prime in an arithmetic progression and the Selberg sieve What is $\tau$ in the last formula? Is it the "number of divisors" function? |
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May 8 |
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The classifying space of a gauge group It is also strange that LaTeX is completely working on my iPhone, but not on PC. |
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May 8 |
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Functional equations Or, are those separate problems? |
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May 8 |
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Functional equations Putting $x=y=z$, you get $2X^2=1$, and $X^3=1$ where $X=f(x,x)$, which is nonsense. |
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May 7 |
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average involving phi function I think so too. |
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May 7 |
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average involving phi function It is also possible to obtain an asymptotic formula with main term $C\frac{1}{\log N}$ for some positive constant $C$. |
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Apr 20 |
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Efficient (divergent) summation for sum of zetas at negative arguments? See this: wolframalpha.com/input/… |
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Apr 20 |
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Efficient (divergent) summation for sum of zetas at negative arguments? That must be an error, this function does have limit as $s\rightarrow 0$, and it is $1+\zeta'(0)$. |
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Apr 19 |
answered | Efficient (divergent) summation for sum of zetas at negative arguments? |
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Apr 19 |
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Efficient (divergent) summation for sum of zetas at negative arguments? I will write proof of $m=-1$ case as an answer. |
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Apr 19 |
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Efficient (divergent) summation for sum of zetas at negative arguments? Using that $\zeta'(0)=-\frac{1}{2}\log 2\pi$, I could prove that the approximation you have is correct. |
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Apr 19 |
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Efficient (divergent) summation for sum of zetas at negative arguments? So, for $m=-1$, did you prove it? or is it just a numerical approximation? |
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Apr 19 |
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When integer polynomials take integer values, does their GCD also take integer values? Yes, but I am pointing out that the GCD should be considered as UFD element. In particular in $\mathbb{Z}[x]$, the GCD of $P$, and $Q$ is, some integer multiple of OP's definition. For example, GCD of $2x^2$ and $8x^2+4x$ is $2x$. |
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Apr 8 |
revised |
Ordinary Generating Function for Mobius added 202 characters in body |
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Apr 8 |
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Ordinary Generating Function for Mobius deleted 204 characters in body |
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Apr 8 |
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Ordinary Generating Function for Mobius added 198 characters in body; added 12 characters in body |
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Apr 8 |
revised |
Ordinary Generating Function for Mobius added 386 characters in body |
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Apr 7 |
revised |
Ordinary Generating Function for Mobius added ; added 30 characters in body; added 11 characters in body; added 10 characters in body; deleted 4 characters in body |
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Apr 6 |
awarded | ● Critic |
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Apr 3 |
asked | Ordinary Generating Function for Mobius |
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Mar 22 |
answered | $\prod_{n=1}^{\infty} n{}^{\mu(n)}=\frac{1}{4 \pi ^2}$ |
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Mar 11 |
answered | Expression for the sum of square roots of zeros of a polynomial |
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Mar 8 |
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Invertibility of a certain matrix indexed by the Hamming cube Added a comment about inverse. |
