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bio website math.lsa.umich.edu/~speyer
location Ann Arbor
age 34
visits member for 4 years, 11 months
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Associate Professor of Mathematics at the University of Michigan. My research interests are in combinatorial algebraic geometry, particularly Schubert calculus, matroids and cluster algebras. I also enjoy thinking about number theory and computational mathematics.


1h
comment Go I Know Not Whither and Fetch I Know Not What
@FelipeVoloch Nonetheless, sorry about that.
2h
comment Go I Know Not Whither and Fetch I Know Not What
No, I didn't do that one. The primes $2$, $3$ and $5$ are going to be wonky since (as GH from MO points out) you aren't working with an integer basis for the ring of integers, and I didn't want to work that hard.
2h
answered Go I Know Not Whither and Fetch I Know Not What
8h
comment Prove that the Dirichlet eta function is monotonic
Whoops! Undergrad, not underaged!
10h
comment Prove that the Dirichlet eta function is monotonic
@Alexander You can group terms of a conditionally convergent sum, you just can't rearrange them. Write out the partial sums of $(1/2) + 1/2 \sum (1/(2k-1)^s - 2/(2k)^s + 1/(2k+1)^s)$ and check that they approach the same limit as the partial sums of $\sum (-1)^{n-1}/n^s$.
10h
comment Prove that the Dirichlet eta function is monotonic
@Alexander Agreed. I think what I wrote is at a good level for communicating with other professionals but you are right, there are subtleties here, since $\sum (-1)^{n-1} e^{-nx}$ isn't uniformly convergent on $[0, \infty)$. If I were presenting this to an underaged analysis class, there would have to be a discussion about cutting the integral into $[a, \infty)$, where the convergence is uniform so we can interchange $\sum$ and $\int$, and $[0,a)$, where the integral is at most $\int_0^a x^s \frac{dx}{x} = a^s/s$. Then send $a \to 0$ to complete the computation.
1d
comment Prove that the Dirichlet eta function is monotonic
@GHfromMO Agreed, rearrangement is the case $g(x)=1$, $s(x) = t(-x)$ (or $s(x) = t(x^{-1})$ if we are using $dx/x$). Chebyshev's inequality is $g(x) = t(x) = 1$.
1d
revised Prove that the Dirichlet eta function is monotonic
added 542 characters in body
1d
awarded  Nice Answer
2d
answered Prove that the Dirichlet eta function is monotonic
2d
comment Prove that the Dirichlet eta function is monotonic
There is an easy proof that $\eta(p) \geq \eta(0) = 1/2$. We have $\eta(p) = (1/2) + \sum_{n=0}^{\infty} (1/2) (2n)^{-p} - (2n+1)^{-p} + (1/2) (2n+2)^{-p} \geq (1/2) \sum_{n=0}^{\infty} \sqrt{(2n)(2n+2)}^{-p} - (2n+1)^{-p} \geq 1/2$ where the first inequality is AMGM. I've been trying to adapt this to show $\eta'(p)>0$, but no luck yet.
2d
awarded  Enlightened
2d
awarded  Nice Answer
Sep
12
awarded  Enlightened
Sep
12
awarded  Nice Answer
Sep
12
answered What is the Implicit Function Theorem good for?
Sep
10
comment Why does the Gamma-function complete the Riemann Zeta function?
If I'm thinking correctly, $\zeta(s)$ is formally $\Gamma(g,1-s)$ where $g$ is a sum of $\delta$ functions at the positive integers. Is there some sense in which $g = \hat{g}$ here?
Sep
10
comment Inequality of arithmetic, geometric and harmonic means
Over at math.SE, we have had two questions about for which $\theta$ (as a function of $n$), we have $GM \leq (1-\theta) AM + \theta HM$. (Your question asks whether $\theta=1/2$ works.) We know that the optimal $\theta$ goes to $0$ as $n \to \infty$, and $\theta=1/n$ works, but there is still some room for improvement. math.SE links: math.stackexchange.com/questions/92935 math.stackexchange.com/questions/803960
Sep
8
comment Solving polynomials of arbitrary degree
Other previous questions mathoverflow.net/questions/89144/… mathoverflow.net/questions/61409/… .
Sep
6
awarded  Popular Question