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Let $K$ be an imaginary quadratic number field (so there are no real embeddings) with ring of integers $\mathcal{O}_K$ . Let $w$ be the number of units in $K$ and $h$ be the class number of $K$. Let $ …

For an integer $k\geq 1$, let $f:\mathbb{C}^k\to\mathbb{C}$ be such that $f$ is analytic in the region $\text{Re}(u_i) > -1$ (say) for each $1\leq i \leq k$, and decays rapidly on vertical lines (i.e. …

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 …

Recently, I've been reading about asymptotics for smooth numbers as well as smooth numbers in arithmetic progressions. One of the ideas I find especially pleasing among some of these results is the us …

I was recently looking into an old problem of Hardy which studies the distribution of integers of the form $2^a 3^b \leq x$, where $a,b\geq 0$. Letting $N(x)$ denote the number of pairs $(a,b)$ satisf …

Let $c>1$, $c\not\in\mathbb{Z}$ and consider the sum $$ \sum_{n\leq x} \tau(\lfloor n^c \rfloor), $$ where $\tau(n)$ is the number of divisors of $n$. I'm almost certain I've seen an evaluation of thi …

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 …

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 …

In their paper "Exponential Sums with Monomials," Fouvry and Iwaniec study exponential sums roughly of the form $$ \sum_{m_1 \sim M_1} \cdots \sum_{m_r \sim M_r} c_1(m_1) \cdots c_r(m_r) e\left(x \fra …

Let $\pi(x;q,a)$ count the number of primes $\leq x$ congruent to $a$ mod $q$. The Brun-Titchmarsh Theorem states that for all $q< x$, $(a,q)=1$, we have $$ \tag{1} \pi(x;q,a) \leq \frac{2x}{\varphi(q …

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\}, …

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 …