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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 …

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 …

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). …

In some of my recent research efforts, I've been applying a lot of estimates for exponential sums involving exponent pairs. Two seemingly simple questions have arisen from these calculations, and I pr …

I realized this is actually pretty much the same problem as one arising when studying squarefree numbers in short intervals. Using some results from Filaseta and Trifonov [1] (specifically the estimat …

Let $f(n)$ be a nonnegative arithmetic function satisfying $f(p^l) \leq A_1^l$ for all primes $p$, integers $l\geq 1$, and some constant $A_1$; $f(n) \leq A_2 n^\varepsilon$ for all $\varepsilon > 0$ …

I'm working on a problem right now in which I need an upper bound for an exponential sum of the form $$ \tag{1} \sum_{N < n \leq 2N} \tau_3(n) e(f(n)), $$ where $\tau_3(n) = \sum_{d_1d_2d_3=n} 1$ and …

A particular case of Dickson's Conjecture states that for $a_1,q_1,a_2,q_2$ with $(a_1,q_1)=(a_2,q_2)=1$, there are infinitely many $n$ for which $q_1 n + a_1$ and $q_2 n+a_2$ are both prime, provided …

This is a bit of a follow up question to this question I asked a couple days ago. The main content of that post can be phrased as asking for a nontrivial lower bound on the sum $$ \sum_{n\leq x} \Lamb …