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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 …

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 …

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 …

In studying the distribution of squarefree numbers in short intervals, Filaseta and Trifonov used some ingenious techniques to obtain various upper bounds for the size of the set $$ S(X) = \left\{u\in …

I'm new to asking questions on MathOverflow, so forgive me if this question is not the kind of thing to be asked here. Let $q$ be a positive integer and let $N$ be an integer with $1 \leq N \leq q$. T …

Let $q$ be a positive integer and $\theta$ a real number with $0 \leq \theta < 1$. Consider the two sums $$ S_\theta^\pm(q)=\sum_{\substack{r=1\\ (r,q)=1}}^{q-1} \left\{\theta\pm\frac{r}{q} \right\}, …

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 $N(\sigma,T)$, …

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 …

A classical result of Fouvry and Iwaniec states that if $\alpha_1,\ldots, \alpha_4$ are nonzero, $M_1,\ldots, M_4 \geq 1$, $X > 0$, and $|\varphi_{m_1,m_2}|,|\psi_{m_3,m_4}|\leq 1$ are complex numbers …

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 …

In a now classical paper, Iwaniec proved the following theorem. Theorem. Let $T \geq 2$, $T^{1/2} < T_0 \leq T$ and $T \leq t_1 < t_2 < \cdots < t_R \leq 2T$, $t_{r+1} - t_r \geq T_0$. Then $$ \tag{1} …

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). …

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 …

Presumably, one could ask this question for a Größencharakter in an arbitrary number field, but I'll restrict my attention to the case I'm interested in. Let $K$ be an imaginary quadratic field with r …

Let $K$ be an algebraic number field, $\mathcal{O}_K$ its ring of integers, $\mathfrak{m}$ an integral ideal of $\mathcal{O}_K$. Let $J$ be the set of all fractional ideals, $P$ the set of principal f …