Anonymous
Reputation
201
Next privilege 250 Rep.
 Feb 29 awarded Yearling Apr 27 comment Bounds for the largest divisor of n less than n^0.5 The state of the art for questions such as this is Kevin Ford's paper arxiv.org/pdf/math/0401223v5.pdf. For the $f(n)$ given in the question, or in fact any $f(n)$ with $f(n) = n^{1/2+o(1)}$, one gets from Ford's work that the set of $n$ with $d(n) \ge f(n)$ has asymptotic density zero. Feb 5 comment Is Euler's totient sub-homogeneous of degree 1? Another way of seeing this: $\phi(mn)$ is the number of integers in $[1,mn]$ coprime to $mn$, while $m\phi(n)$ is the number of integers in $[1,mn]$ coprime to $n$. Jan 1 comment 3-7 primes in base 10 I will add that almost-prime'' results of this sort have been obtained by Dartyge and Mauduit: Almost primes whose expansion in base $r$ misses some digits. (Nombres presque premiers dont l'écriture en base $r$ ne comporte pas certains chiffres.) (French) J. Number Theory 81, No.2, 270-291 (2000) Jan 1 comment 3-7 primes in base 10 Some questions of a similar flavor, for the much denser set of squarefree numbers replacing the set of primes, are considered by Filaseta and Konyagin in this Acta Arithmetica paper: matwbn.icm.edu.pl/ksiazki/aa/aa74/aa7431.pdf In particular, they prove that for each $2 \leq b \leq 5$, there are infinitely many squarefree numbers in base $b$ consisting only of the digits $0$ and $1$. When $b=10$, they can show there are infinitely many squarefree numbers consisting just of the digits $0$, $1$, and $2$. Dec 5 comment Average orders of multiplicative functions Chapter IV of this book is a compendium of results related to four non-multiplicative functions (the largest prime divisor, the smallest prime divisor, and the sum of the prime divisors, counted with or without multiplicity). Now there may still be a result in this chapter that fits the bill, but maybe a more precise reference could be given? Nov 17 awarded Enlightened Nov 17 awarded Nice Answer Nov 16 comment Least prime primitive root It looks from Wang's paper (which can be found by searching Google books for his collected works) that he actually shows the partial sums of $\Lambda(n) e^{-n/x}$, taken over primitive roots $n$ up to about $(log p)^{C}$, is positive. So this gives a prime small power $n$ which is a primitive root mod $p$. But then the prime which $n$ is a power of must also be a primitive root mod $p$. Nov 16 awarded Editor Nov 16 revised Least prime primitive root corrected e^{-\gamma} to e^{\gamma} Nov 16 comment Least prime primitive root OK, answer posted! Nov 16 answered Least prime primitive root Nov 10 awarded Necromancer Nov 10 awarded Teacher Nov 10 answered Question on consecutive integers with similar prime factorizations