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Nov
18 |
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Proving Legendre's Sum of 3 Squares Theorem via Geometry of Numbers
Have you seen this recent arXiv preprint? arxiv.org/abs/1509.02590 |
Nov
8 |
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What is the asymptotic growth rate of the product of divisor function up to n
Well, $f(n)=\log\tau(n)$ is an additive function; that is, $f(n) = \sum_{p^k\parallel n} f(p^k)$. Now insert this expression for $f$ into $\sum_{n \le x}f(n)$ and reverse the order of summation. I believe in this example, one finds that the main term in the asymptotic is $x \sum_{p \le x} f(p)/p$, and this is $\sim (\log 2) x\log\log{x}$, as $x\to \infty$ (Alternatively, note that $\tau(n)$ is between $2^{\omega(n)}$ and $2^{\Omega(n)}$, and use the known results --- as found in Hardy and Wright, for example --- on the partial sums of $\omega$ and $\Omega$.) |
Oct
31 |
answered | What is known about the largest prime divisor of the product of $k$ consecutive integers? |
Oct
25 |
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multiplicative functions of powers
There's a beautiful expository paper by Moree and Cazaran that touches on this; see cl.ly/0V012j0O1a1B |
Oct
25 |
answered | multiplicative functions of powers |
Oct
25 |
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multiplicative functions of powers
Perhaps I don't understand the question. But $\tau(n^k)$ is a nonnegative multiplicative function. And there are very general theorems giving asymptotics for partial sums of nonnegative multiplicative functions; e.g., there is a beautiful theorem of Wirsing that immediately applies to this question. See Wirsing, Eduard Das asymptotische Verhalten von Summen über multiplikative Funktionen. (German) Math. Ann. 143 1961 75–102. (But I think Wirsing is "overkill" here, in the sense that Brad Rodgers's suggestion above also works and is simpler.) |
Oct
19 |
answered | How often is $2^n-1$ a number with few divisors? |
Oct
5 |
answered | Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$? |
Sep
28 |
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Does the equation $241+2^{2s+1}=m^2$ have a solution?
If anyone is wondering, the argument Mike Bennett refers to is worked out in math.dartmouth.edu/~carlp/2tokmmminus1v8.pdf . See Theorem 1. |
Aug
15 |
answered | Prime divisors of values of a polynomial on an infinite set |
Jul
31 |
answered | Lower bounding the probability that $\gcd(t,N)≤B$, for a random $t$ and fixed (large) $N$ |
Jul
28 |
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Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial
An alternative is to use some of the theory of Beurling primes. See alpha.math.uga.edu/~pollack/beurling.pdf |
Jul
28 |
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Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial
A quick comment: One can derive the estimate $\sum_{p \le x} \nu_F(p)/p = \log\log{x} + O(1)$ without worrying about zeros of $\zeta_K(s)$ on $\Re(s)=1$; e.g., apply the Tauberian theorem mentioned as Proposition 5 in alpha.math.uga.edu/~pollack/eulerprime.pdf to $\log \zeta_K(s)$. Everything needed to check the hypotheses is in Hecke's algebraic number theory book (and goes back to Dedekind, I think). |
Jul
17 |
awarded | Yearling |
Jun
25 |
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Number of primes one larger than divisors of a fixed number, which is LCM of 1,2,3,…,L
I don't have an account on Math Stackexchange at the moment, but the conjecture of Wythagoras that motivated your question follows quickly from Theorem 1 in this paper of Erdos--Pomerance--Schmutz: math.drexel.edu/~eschmutz/PAPERS/lambda.pdf |
Jun
6 |
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When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$?
Have you seen Proposition 3.8 and Example 3.16 in esc.fnwi.uva.nl/thesis/centraal/files/f310232185.pdf ? |
May
25 |
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On the number of consecutive divisors of an integer
Here's a PDF: renyi.hu/~p_erdos/1978-26.pdf |
May
25 |
answered | On the number of consecutive divisors of an integer |
May
14 |
answered | On the natural density of almost perfect numbers |
May
10 |
revised |
Are There Infinitly Many $n$ Which $a\times n!+1$ Be Composite?
deleted extraneous plus sign |