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Sep 19 |
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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 |
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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 |
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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 |
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Is there a von Koch-type theorem for the generalized Riemann hypothesis?
@GHfromMO: Thanks; I didn't know that before! |

Sep 18 |
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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 |
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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 |
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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 |
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$x^2+1$ 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 |
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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 |
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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 |
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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 |
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On the maximum number of $t$-subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections
See mathoverflow.net/questions/161159/… |

Sep 12 |
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Can a sum of roots of unity be an integer?
Certainly missed that! I'm curious how you came across this problem. |

Sep 9 |
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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 |
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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 |
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Can a sum of roots of unity be an integer?
That's very nice! |

Sep 5 |
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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 |
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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 |
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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 |
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$L^2$ 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. |