# Sanchez

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I really appreciate it when you take time to answer my questions. Thanks!

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 Feb28 comment How prove this $arg((1+ia)(2+ia)(3+ia)\cdots(n+ia))=\arctan{\frac{a}{1}}+\arctan{\frac{a}{2}}+\cdots+\arctan{\frac{a}{n}}$ You already saw that it's not even true for $n=1$.. Feb21 comment If $x,y,z\in(0;1)$, prove that $(x+1)(y+1)(z+1)\ge \sqrt{8(x+y)(y+z)(z+x)}$. $(x+1)(y+1) \ge 2(x+y) \Leftrightarrow (1-x)(1-y) \ge 0$. Feb17 comment E: $y^2+y=x^3$ an elliptic curve over $F_{2}$. How to prove the number of $E(F_{2^n})$ = $2^n+1$ if n is odd, … @AdamStaples, I think the zeta function solution is exactly the one you posted - the proof of the theorem has Hasse-Weil L function lurking behind. On the other hand, you can also proceed by Jacobi sum. The case where $n$ is even is already done by Jyrki Lahtonen above, and the case where $n$ involves a Jacobi sum for the cubic characters. Feb12 comment class field theory via schemes? mathoverflow.net/questions/73054/… Feb12 comment Why are de Rham cohomology and Cech cohomology of the constant sheaf the same You are right.. Feb6 comment How prove $\sum_{cyc}(f(x^3)+f(xyz)-f(x^2y)-f(x^2z))\ge 0$ Actually, using $1/(1-x) = 1+x+x^2+...$ + schur should work already. Feb1 answered Is there a general pattern behind the decimal expansion of $\frac{1}{7}$ being $.14+.0028+.000056+.00000112+…=.\overline{142857}$? Feb1 comment How prove $\sum_{cyc}(f(x^3)+f(xyz)-f(x^2y)-f(x^2z))\ge 0$ See artofproblemsolving.com/Forum/viewtopic.php?t=22364?ml=1 post #4. Your problem is a special case of a general form of Popoviciu's inequality, since $\frac{1}{1-e^x}$ is a convex function in your range. Using Vasc's inequality in that post, you get automatically that $$f(x^3)+f(y^3)+f(z^3) + \frac{3}{2} f(xyz) \ge \frac{3}{4} (f(x^2y) + \cdots + f(zx^2))$$ This is weaker than what you are proposing, but since your $f$ is special it's quite possible to wiggle the proof of that inequaity (which is for convex functions) for this case. Jan29 comment What does the term “distinguished basis” mean? "Distinguished" generally means something stand-out. There are many basis for a vector space, but in familiar ones there are usually a "natural" one that occurs. In $\mathbb{R}^n$ it could be the standard basis $\{e_1,\cdots,e_n\}$. In polynomials it could be $1,x,x^2,\cdots$. Those would be reasonably called "distinguished basis". Jan25 comment Min $A=14(a^{2}+b^{2}+c^{2})+\frac{ab+bc+ac}{a^{2}b+b^{2}c+c^{2}a}$? Just came across this - what next? To me it's very unlikely that bounding the first term like that would work, because the second term has an infimum of 1 and that is not achieved when the three variables are equal. Jan19 comment How prove this inequality $\sum_{cyc}\frac{a^2}{b(a^2-ab+b^2)}\ge\frac{9}{a+b+c}$ @EwanDelanoy, he only assumes $a$ is the minimum - doesn't seem to assume $u \ge v$ or vice versa. Jan19 comment How prove this inequality $\sum_{cyc}\frac{a^2}{b(a^2-ab+b^2)}\ge\frac{9}{a+b+c}$ By the way, there's no problem to assume $a+b+c = 3$ at all. Homogenity argument allows one to assume $a+b+c = C$ for any positive C. Jan15 comment An inequality with $a,b,c>0$ The sign is reversed. Jan10 comment Riemannian geometry: …Why is it called 'Exponential' map? In the case of matrix groups, it really is the matrix exponential. Jan6 awarded Pundit Jan6 comment What is $Spec(\mathbb{Z}[x])$? Try to find the treasure map laid out by Mumford. (If you don't know what this means, try to google) Jan2 comment Gross-Zagier formulae outside of number theory Nice question! Perhaps you should post this to MO? Jan2 comment zeros about Riemann Zeta function and some L-function It is, but for Dirichlet L functions the only interesting case for non-existence of Siegel zeros is for real (thus quadratic) nontrivial character. Jan1 comment Examples of non-isomorphic fields with isomorphic group of units and additive group structure $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$? Dec31 comment on the generating series $\sum_{n\geq 1}\frac{\sigma_a(n)}{n}x^n$ Is there any reason you think that the power series has a nice expression?