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Sep
19
comment Number of solutions in a sum of squares Diophantine equation
For fixed $p\ge 3$ the answer is correct and the bound is tight, as you note. But for $p=2$ you need an extra $\log n$ factor.
Sep
19
comment Inequality due to Siegel (assumptions) and upper bounds on number field discriminants
The inequality in (*) should give a lower bound for $d$ rather than an upper bound. It looks like you're interested in lower bounds for discriminants. Look up Odlyzko's survey article on this (available from his website).
Sep
19
comment Is there a von Koch-type theorem for the generalized Riemann hypothesis?
@KConrad: Yes the last para is for a fixed $q$. But note that $\psi(x;q,1)$ is being compared to $\psi(x)$ rather than $x$. So in this formulation I can get away without assuming RH.
Sep
19
comment Is there a von Koch-type theorem for the generalized Riemann hypothesis?
@GHfromMO: Thanks; I didn't know that before!
Sep
18
comment What are all positive integers n for which the congruence $a^{n+1} \equiv a (mod n)$ holds?
The edited question makes all the earlier comments and answers incomprehensible. This is not a good way to proceed.
Sep
18
comment Are major arcs always around a fraction with small denominator?
Not a very well posed question. E.g. if $A$ is the set of numbers for which $n\sqrt{2}$ has fractional part less than $1/100$ then the exponential sum will be large near $\sqrt{2}$ (assuming here $a_n=1$ on that set).
Sep
16
comment Squarefree Parts of Mersenne Numbers with prime exponent
The question is unclear. But there is a remarkable conjecture that $2^p-1$ is always squarefree for prime numbers $p$. The point is that if $2^p-1$ is divisible by $\ell^2$ for some prime $\ell$, then $\ell$ must be Wieferich, and the order of $2\pmod \ell$ would also have to prime (namely $p$). Wieferich primes are themselves rare (only two are known), and those with the order of $2$ being prime should be so rare as not to exist!
Sep
15
comment $x^2+1$ attaining almost prime values
Most of the numbers with $r$ prime factors have $r-1$ small prime factors and one large prime factor. Given that, I think it would be difficult to evaluate $N_r(x)$, since eventually you will be reduced to estimating primes in a quadratic sequence. I don't quite understand your last sentence: any sieve method will give an upper bound of the right order. Maybe you mean right constant as well? Finally there's a difference between counting numbers with $r$ prime factors, and weighting with $\Lambda_r(n)$. My comment applies to the first form (the question as written) and not the second.
Sep
14
comment Solving a non linear equation
If $K$ is reasonably large then the fixed point seems very close to $K^{-2/3}$ -- even for $K=6$ this is a pretty close approximation. Some asymptotic analysis around here (which might be unpleasant but straightforward) should settle the issue -- first show that the fixed points have to be reasonably close to $K^{-2/3}$ and then use the derivative to check uniqueness.
Sep
14
comment On the maximum number of $t$-subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections
In your statement of Wilson's theorem don't you need the condition that $t-1$ divides $n-1$?
Sep
13
comment On the maximum number of $t$-subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections
@DouglasZare: Actually my answer there gave an explicit upper bound which is tight in some cases.
Sep
13
comment On the maximum number of $t$-subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections
See mathoverflow.net/questions/161159/…
Sep
12
comment Can a sum of roots of unity be an integer?
Certainly missed that! I'm curious how you came across this problem.
Sep
9
comment The maximum of the preimage of [1,x] through Euler's totient function
Choose the largest primorial below $\sqrt{x}$, and find a multiple of this close to $e^{\gamma} x\log \log x$. That does the job (together with the classical lower bound for $\phi(n)$).
Sep
9
comment What is the status of the Gauss Circle Problem?
Today's arXiv posting by Shaneson arxiv.org/pdf/1409.2446.pdf indicates that he and Cappell were unable to find an ``error-free version" of their announced proof.
Sep
9
comment Can a sum of roots of unity be an integer?
That's very nice!
Sep
5
comment Can a sum of roots of unity be an integer?
The Evans paper is interesting, but I haven's studied it fully yet. I also found a simple proof for all prime powers, and will write this up maybe later today.
Sep
5
comment Laplacian eigenfunction $L^p$ norms
I haven't read this paper myself, but I believe the work of Sogge (J. Funct. Analysis 1988) may have results of the kind you want?
Sep
4
comment Can a sum of roots of unity be an integer?
@GNiklasch: The question is a bit unclear to me. It seems to want integrality for every $k$ dividing $n$ in a certain range (but maybe this is just an unclear formulation). And, $k=n$ is permitted and seems to impose already a very strong restriction.
Sep
4
comment $L^2$ discrepancy bound for sequences in $[0,1)$
A few points: one is that Davenport needs to consider $\{n\alpha\}$ for positive and negative $n$. This is easily fixed by considering $x_{2n} = \{n\sqrt{2}\}$ and $x_{2n+1} = \{-n\sqrt{2}\}$ say. So you could try computing these. Note that only an upper bound is established, and not an asymptotic -- there will be fluctuations. The log arises from rational approximations to $\sqrt{2}$, there are constants involved, and also the denominators of the convergents grow exponentially. Numerical computations up to $32$ may not be very insightful; I recommend reading the (simple) proof.