1,833 research outputs found
Bounded gaps between primes with a given primitive root
Fix an integer that is not a perfect square. In 1927, Artin
conjectured that there are infinitely many primes for which is a primitive
root. Forty years later, Hooley showed that Artin's conjecture follows from the
Generalized Riemann Hypothesis (GRH). We inject Hooley's analysis into the
Maynard--Tao work on bounded gaps between primes. This leads to the following
GRH-conditional result: Fix an integer . If
is the sequence of primes possessing as a primitive root, then
, where is a finite
constant that depends on but not on . We also show that the primes in this result may be taken to be consecutive.Comment: small corrections to the treatment of \sum_1 on pp. 11--1
On the degrees of divisors of T^n-1
Fix a field . In this paper, we study the sets \D_F(n) \subset [0,n]
defined by [\D_F(n):= {0 \leq m \leq n: T^n-1\text{has a divisor of degree
in} F[T]}.] When \D_F(n) consists of all integers with ,
so that has a divisor of every degree, we call an -practical
number. The terminology here is suggested by an analogy with the practical
numbers of Srinivasan, which are numbers for which every integer can be written as a sum of distinct divisors of . Our first
theorem states that, for any number field and any ,
[#{\text{-practical }} \asymp_{F} \frac{x}{\log{x}};] this extends
work of the second author, who obtained this estimate when F=\Q.
Suppose now that , and let be a natural number in . We
ask: For how many does belong to \D_F(n)? We prove upper
bounds in this problem for both F=\Q and F=\F_p (with prime), the
latter conditional on the Generalized Riemann Hypothesis. In both cases, we
find that the number of such is ,
uniformly in
Arithmetic functions at consecutive shifted primes
For each of the functions and every
natural number , we show that there are infinitely many solutions to the
inequalities , and similarly
for . We also answer some
questions of Sierpi\'nski on the digit sums of consecutive primes. The
arguments make essential use of Maynard and Tao's method for producing many
primes in intervals of bounded length.Comment: Made some improvements in the organization and expositio
Uncertainty principles connected with the M\"{o}bius inversion formula
We say that two arithmetic functions f and g form a Mobius pair if f(n) =
\sum_{d \mid n} g(d) for all natural numbers n. In that case, g can be
expressed in terms of f by the familiar Mobius inversion formula of elementary
number theory. In a previous paper, the first-named author showed that if the
members f and g of a Mobius pair are both finitely supported, then both
functions vanish identically. Here we prove two significantly stronger versions
of this uncertainty principle. A corollary is that in a nonzero Mobius pair,
either \sum_{n \in supp(f)} 1/n or \sum_{n \in supp(g)} 1/n diverges.Comment: 10 page
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