Sanchez

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

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 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? Dec23 comment If $abc=1$ and $a,b,c$ are positive real numbers, prove that ${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1$. @mathh, it's true here, by AM-GM inequality, that $$a+b+c \ge 3$$. Dec18 comment How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$ $A \ge \frac{1}{2}$ as shown in Greg Martin's answer. In particular, $1/\sqrt{10}$ can't be right. Dec15 comment Hartshorne's definition of structure sheaf disjoint union, or coproduct in the category of sets. Dec14 comment If $abc=1$ and $a,b,c$ are positive real numbers, prove that ${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1$. @mathh, if you know Muirhead's inequality, at the final step you just homogenize your inequality (i.e. multiply $(abc)^{2/3}$ to $a+b+c$ part) and use Muirhead. If not, then try to find a clever AM-GM: for example: $a^2b+a^2c \ge 2a^{3/2}$ by AM-GM, then you can use Power mean or other approaches to prove $a^{3/2} + b^{3/2} + c^{3/2} \ge a+b+c$. Dec14 answered If $abc=1$ and $a,b,c$ are positive real numbers, prove that ${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1$. Dec11 comment Why is this result of the Cauchy-Goursat theorem true? You can also write down the anti-derivative like John said, as long as $n \neq 1$. So if you can see that for polynomials, you should be able to see that for $n \ge 2$ too. As for $n = 1$, interpretation of integrating $dz/z$ as winding number is probably helpful. Dec11 comment Why is this result of the Cauchy-Goursat theorem true? what makes you think that intuitively, integrating polynomial over line is 0? Dec5 comment An exercise in Silverman You are working in $\mathbb{P}^2$ I believe, which means that $[x,y,z] = [tx,ty,tz]$. In particular if $z \neq 0$, you can normalize it to make it 1, i.e. you can always plug in $z = 0$ or $z=1$ in your equations. Dec5 comment Trying to show AM-GM mean inequality It is. It's generally called the weighted AM-GM inequality. Dec5 answered How to find the minimum of $a+b+\sqrt{a^2+b^2}$ Dec5 comment How to find the minimum of $a+b+\sqrt{a^2+b^2}$ Are you forbidding calculus? Dec4 comment How prove this inequality $\sin{\sin{\sin{\sin{x}}}}\le\frac{4}{5}\cos{\cos{\cos{\cos{x}}}}$ How did you find 4/5? Dec4 comment What is the value of $\sum_{i=1}^n X_i(g) X_i(h^{-1})$ when $g,h \in G$ are in the same conjugacy class? Or more simply, note that it's the sum of the trace of $g$ acting on $Hom(\rho_i, \rho_i)$ if $\rho_i$ accounts for all irrep of $G$. But $\oplus Hom(\rho_i, \rho_i) = \mathbb{C}[G]$, so you are looking for the trace for $g$ acting on the group algebra. This is $|G|$ if $g=1$ and 0 otherwise. Dec4 comment Dimension of the tangent space of an algebraic set Your $k$ is not $\mathbb{R}$ (which isn't even algebraically closed!). So your (a) and (c) have problems. Dec4 comment Is there an elementary proof that there are infinitely many primes p satisfying the following: It doesn't feel like Chebotarev is necessary - Dirichlet's theorem on arithmetic progression should already be enough, if you are willing to get your hands dirty to show directly for instance 2 is a QR but not a fourth root mod p iff p satisfies some congruence conditions, which would of course follows from biquadratic reciprocity. Dec3 comment Arithmetic in p-adic You don't need Hensel's lemma to know that there is no solution in $\mathbb{Z}_5$: The natural homomorphism $\mathbb{Z}_5 \to \mathbb{F}_5$ shows that if there is a squareroot of 2 in $\mathbb{Z}_5$, then there is one in $\mathbb{F}_5$, and you know that the latter is false. It's the converse where Hensel's came in, and as Luis said, you will probably just prove Hensel's on the way without realizing it. I do think this is the only way of doing it. Dec3 comment Is there an elementary proof that there are infinitely many primes p satisfying the following: It seems that biquadratic reciprocity has to come in somewhere - is there any reason you think that an elementary proof exists?