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Jul
10
comment Asymptotic expansion of a sequence given by an integral with reciprocal Gamma function
Yes, that's right -- $1/(\log n)^2$.
Jul
9
comment Asymptotic expansion of a sequence given by an integral with reciprocal Gamma function
It's about $n/(\log n)^2$. The integral is dominated by the region near $1$, where you can use $-1/\Gamma(x-1)=(1-x) + O((1-x)^2)$ ...
Jul
7
comment Explicit bounds for exceptional zeros and/or $L(1,\chi)$ for real $\chi$
Yes; the results should only get better for imprimitive characters. Also the $\log^2$ in the denominator can be removed while still using only the trivial bound for the class number (but maybe no one wrote up an explicit version of that -- maybe useful to do if you're really fighting for constants) -- see Goldfeld and Schinzel archive.numdam.org/ARCHIVE/ASNSP/ASNSP_1975_4_2_4/…
Jul
7
comment Explicit bounds for exceptional zeros and/or $L(1,\chi)$ for real $\chi$
The (standard) argument in Ford/Luca/Moree doesn't really use that $q$ is prime, and works for any fundamental discriminant.
Jul
1
comment Number of prime numbers in a range
No, thanks to the recent breakthroughs in bounded gaps between primes.
Jun
30
comment One-to-one correspondance between zeta zeros and the prime powers?
This may have something interesting, but there is no clear question as it stands. The explicit formula of course gives a correspondence between zeta zeros and primes, but it's not clear what exactly you're after here.
Jun
29
comment Probability that random nonnegative integer matrix is singular
For fixed $n$ and large $k$, the probability is known to go to zero. See for example this paper of Martin and Wong which contains more references: math.ubc.ca/~gerg/papers/downloads/AAIMHNIE.pdf . Also, the work of Rudelson and Vershynin arxiv.org/pdf/math/0703503.pdf which gives very general situations where the probability goes to zero.
Jun
13
comment What is the analytic conductor of this Hecke L-function?
Use the duplication formula for the Gamma function which connects $\Gamma(2s)$ to $\Gamma(s) \Gamma(s+1/2)$.
Jun
5
comment Euler-like identity for partition function
Relevant: mathcs.emory.edu/~ono/publications-cv/pdfs/017.pdf
Jun
5
comment Minimal Discriminants
Odlyzko mentions this problem in his survey paper dtc.umn.edu/~odlyzko/doc/arch/discriminant.survey.pdf asking whether the root discriminant goes to infinity for prime degrees (see open problem 2.4). I don't think much more is known about it, although I would love to be wrong here!
Jun
4
comment $L^1$ norm of exponential sum of $n^2 x$
@tdw: Dear Trevor, From the works ofJurkat and van Horne and Marklof, the quadratic Weyl sums have a distribution that is not Gaussian. So the constant in the moments, I don't think needs to match your conjecture. The constant they get is by averaging moments of a theta function over a fundamental domain. It is possible that for the first moment this could evaluate to your conjectured value, but I don't see why. In any case, the distribution is not Gaussian, which seems quite different from other powers. Am I missing something?
Jun
4
comment Approximate homomorphisms
I was thinking of the ideas around Corollary 2.4 in that paper. (And also that ${\Bbb Z}/p$ for large enough $p$ might behave not too differently from ${\Bbb Z}$.) Anyway, just a quick thought.
Jun
4
comment Approximate homomorphisms
The techniques in this paper arxiv.org/pdf/1308.2247v1.pdf might be useful.
May
28
comment The sum of squared logarithms conjecture
@JohannesHahn: I'm surprised by your comment. The question clearly states a conjecture and asks for a proof -- what's not on topic about this? And, anyone who doesn't want the gold can pass it on to me!
May
26
comment Radial limit does not exist almost everywhere
@MattYoung: Slightly more general version -- lacunary Fourier series with coefficients whose $\ell^2$ norm is infinite. Is that what you looked at, or with coefficients being $1$?
May
26
comment Radial limit does not exist almost everywhere
@GHfromMO: Thanks for pointing that out -- I'll correct it. For the second part, my idea would be to compute moments on the circle with radius $r$. Since the powers of $2$ don't have too many linear relations, one would get an approximately (complex) Gaussian with variance tending to infinity (this is because of the assumption on coefficients). Then it follows that on each circle of radius sufficiently close to $1$, most points (measure close to $1$) will have a large value of the function. From that the result follows.
May
26
comment On the number of consecutive divisors of an integer
@CaptainDarling: Note it's $\exp(-(\log z)^{2+o(1)})$, not like $\exp(-z^2/2)$ (Gaussian). I don't expect it to be anything nice.
May
26
comment Radial limit does not exist almost everywhere
Note that almost every real number $\theta$ will have the property that its binary expansion will have arbitrarily long strings of zeros. Now suppose $N$ is such that $\Vert 2^N \theta \Vert \le 2^{-h}$, and consider $f(e^{-1/2^N} e^{2\pi i\theta})$ and $f(e^{-1/2^{N+h}} e^{2\pi i \theta})$. They should differ by $\gg h$, showing that radial limits don't exist.
May
19
comment Non-standard Gauss sums
Yes indeed! I forgot about my comment!
May
19
comment Non-standard Gauss sums
Isn't Elkies's answer about ruling out the maximal size $2\sqrt{p}$ for Kloosterman sums, rather than zero?