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comment Are there open problems for primes which are known for probable primes?
I thought you had in mind questions that "are entirely different" from the membership question: e.g. twin primes, Goldbach etc. For these questions, usually we have guesses for the answer (e.g. number of solutions to a, a+2 with a in some set) based on the size of the sets in question. My point is that the set of pseudoprimes is so small that this wouldn't affect such heuristics (and indeed it would be much harder for some such property to hold with one pseudoprime involved).
Aug
26
comment Bounds on sum of reciprocal of logarithm of primes
Just use the prime number theorem and partial summation.
Aug
23
comment Maximum norm of discrete Fourier transform
Didn't you ask exactly the same question previously? mathoverflow.net/questions/210836/… It would be better to bump that question up, rather than duplicate it here.
Aug
16
comment Bateman-Horn conjecture, continued
The Galois group acts transitively on the roots, and Burnside's lemma gives that the average number of fixed points is the size of the orbit which is one.
Aug
14
comment Table of LCM's vs. table of products
I think there'll be about as many numbers of the form lcm$(a,b)$. Ford counts integers having a divisor in an interval $[y,2y]$, and one should be able to adapt this to counting square-free integers with such a divisor. Of course, a square-free integer in the multiplication table arises also as lcm$(a,b)$.
Aug
5
comment Polynomials vanishing modulo some integer $n$
A bit unclear to me what you are looking for. Suppose $p$ and $q$ are primes with say $q$ of size about $p^2$. Then $(t^p-t)q$ is a polynomial that is zero mod $n=pq$, and the coefficients are of size at most $q=o(n)$, and the degree is $p = o(p+q)$. Does that qualify as a counterexample?
Aug
3
comment Egyptian fractions similar to Erdos-Straus conjecture
This is an open problem, as Elsholtz comments in ams.org/journals/tran/2001-353-08/S0002-9947-01-02782-9/… (see the second para of the second page of the article).
Jul
31
comment Finding a lower bound in terms of given integers
@BorisBukh: Usually for Baker type methods to work, one fixes some algebraic irrational, and then considers approximations of suitably large height. I don't immediately see how that leads to anything better than Fedor Petrov's comment above to the problem at hand (in the general situation where $n$, $m$ and $l$ may all be large).
Jul
29
comment Fermat's proof for $x^3-y^2=2$
There may be more than one way of writing numbers in the form $a^2+2b^2$. So the last para doesn't seem clear to me.
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?