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God geometrizes continually. But I prefer doing number theory.


Apr
11
awarded  Enlightened
Apr
11
awarded  Nice Answer
Apr
11
revised Roots of unity near 1 in Z / p Z
removed unnecessary use of a dyadic interval
Apr
11
answered Roots of unity near 1 in Z / p Z
Apr
11
comment Roots of unity near 1 in Z / p Z
Suppose $r=3$, so that we are looking at roots of $x^2+x+1$. Then this seems like it should follow immediately from results on uniform distribution of solutions to quadratic congruences mod $p$; see imrn.oxfordjournals.org/content/2000/14/719.abstract
Mar
26
answered Integers up to n having an even number of trailing zeros in their factorial
Mar
6
awarded  Critic
Feb
27
comment Solutions of $rad(\sigma(m))=2rad(m)$
It sounds like you will find the (very similar) problems considered in this paper of interest: math.uga.edu/~pollack/pperfs16.pdf
Jan
10
comment About sign changes of Li(x)-π(x)
Are you familiar with the results of Kaczorowski? See Theorem 1 of matwbn.icm.edu.pl/ksiazki/aa/aa45/aa4517.pdf
Jan
10
answered Generalization of Euler four square formula?
Jan
10
awarded  nt.number-theory
Jan
9
answered Representing primes explicitly with binary quadratic forms
Jan
9
comment Are primes of density 0 in $a\cdot b^n+c$?
Not as far as I know. I would love to be wrong and would even be happy to see a proof that was conditional on just GRH.
Jan
9
awarded  Enlightened
Jan
9
comment For which types of problems can one expect to use Bombieri-Vinogradov in place of GRH?
As you say, B--V is often called "GRH on average". The reason is this: First, think of GRH as a statement about the error term for primes in progressions. B--V says (roughly) that a certain sum of error terms is just as small as it would be if each individual term obeys the GRH-conditional upper bound. In many problems in analytic number theory, the error term at the end of the day is bounded by a sum of error terms for the PNT in AP. For example, this is the case in a lot of sieve problems. Often, such a sum is close enough to the sum estimated by B--V that one can cite it directly.
Jan
9
awarded  Nice Answer
Jan
8
answered Are primes of density 0 in $a\cdot b^n+c$?
Jan
8
revised Proportion of square-free integers $n$ with $\gcd(n,\varphi(n))$ a prime
Changed $x$ to $N$ to agree with the notation of the original question
Jan
8
answered Proportion of square-free integers $n$ with $\gcd(n,\varphi(n))$ a prime
Jan
2
comment The implicit constant in GRH
Bach and Shallit give this reference: J. Oesterlé, Versions effectives du théorème de Chebotarev sous l'hypothèse de Riemann généralisé, Astérisque 61 (1979). Unfortunately, I think this is only an announcement, and that the promised proofs never appeared in print.