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Results tagged with nt.number-theory
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user 307675
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
7
votes
Raising positive integer to $c\in\mathbb{R}-\mathbb{N}$ rarely gives an integer!
To expand on a comment of Lucia, when $c$ is irrational, we can show that there are at most $O((\log N)^2)$ values of $n\leq N$ such that $N^c$ is rational, let alone an integer.
Let $\mathcal{A}$ be …
- 1,778
4
votes
Accepted
Upper bound of number of prime factors
To reiterate Stanley Yao Xiao's comment, one should not expect that $p-1$ has any more or less prime factors than a typical integer of size $p$. For an arbitrary integer $n$, the bound
$$
\omega(n) \l …
- 1,778
6
votes
Iwaniec & Kowalski partial sums of multiplicative functions
For your second question, I think you're correct in noting that they are incorrectly applying their result. I'm sure some sort of result for the sum of squares characteristic function holds, but it do …
- 1,778
7
votes
Accepted
On the nearest integer to $\zeta(1-1/B),B \ge 2$
We can make the error mentioned by Wojowu in his comment explicit by using some results on the Laurent coefficients of the zeta function. There are a few results on this, but I'll just use Theorem 2 o …
- 1,778
11
votes
Density of fake zeros of Zeta
To add to GH from MO's answer, Chapter 10 of Iwaniec and Kowalski's "Analytic Number Theory" is another good reference. A few of the classical papers are
Ingham's "On the Estimation of $N(\sigma,T)$, …
- 1,778
13
votes
About an asymptotic behavior in number theory
See this paper of Naslund, specifically Proposition 3. That result, along with the prime number theorem, shows that
$$
\frac{1}{\pi(N)} \sum_{p\leq N} \left\{ \frac{N}{p} \right\} \sim 1-\gamma,
$$
wh …
- 1,778
10
votes
Accepted
Large values of $\zeta(1/2+it)$ from sums of short moments
I wrestled with this question for an annoyingly long time. Now understanding things much better, I'm a bit embarrassed that this took me so long to figure out (such is the process of learning though). …
- 1,778
6
votes
Approximation of partial sum over prime omega function
By partial summation, one has
$$
S(N) = N\sum_{n=1}^N \omega(n) - \int_{1}^N \bigg(\sum_{n\leq t} \omega(n)\bigg) dt.
$$
Using Mertens' theorem with the classical error term in the prime number theore …
- 1,778
7
votes
Reference for zero sum estimates of Dirichlet L functions
This follows from Noam's answer and the classical zero-free region for Dirichlet $L$-functions. For a specific reference, see Davenport's Multiplicative Number Theory, Chapter 14. The result quoted be …
- 1,778
1
vote
Non-trivial upper bound for a sum related to $p^{-1}z \pmod q$ and $q^{-1}z \pmod p$
This is not mean to be a full answer, but one which illustrates how one can prove an estimate like the one in my comment through a rather ``brute force'' approach.
To illustrate the idea of the comput …
- 1,778
1
vote
Accepted
Mean value of the divisor function over Piatetski-Shapiro sequences
Since asking my question, I have stumbled upon the answer myself, so I post it here in case some future person finds this post.
It appears that the only paper that explicitly considers the problem abo …
- 1,778