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Would it be possible to use the circle method to prove that there are no solutions to certain diophantine equations. For example, could one use the circle method to prove the fact that there are no so …

Most improvements on the zero-free region for $\zeta(s)$ go through bounds on the exponential sums $$\sum_{n\sim N} n^{it}$$ for $N$ in certain ranges depending on $|t|$. Is there any way to directly …

Does anyone know where one might locate "Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam, Volume 2"? There exists Volume 1 here: https://link.springer.com/book/10.1007 …

Is there a quick way to compute (somewhat accurately) for large $X$, the following exponential sum, where $\Lambda$ is the von Mangoldt function? $$\int_0^1 \bigg|\sum_{n\le X} \Lambda(n)e(n\alpha)\bi …

Let $X\ge 2$ be large. Let $$A = \left\{\frac{a}{q}: 1\le a < q\le X,\ (a, q) = 1\right\},$$ and let $$B = \left\{\frac{a}{p}: 1\le a < p\le X,\ (a, p) = 1,\text{ and $p$ is prime}\right\}.$$ Note tha …

Suppose that $H = H(X)$ is some quantity growing with $X$. Are there any bounds on $$F(X, H) = \min_{X < n\le X + H} \omega(n)?$$ It isn't hard to obtain a lower bound $\max_{x\sim X} F(X, H)\gg \frac …

Let $c_q(n)$ be the Ramanujan sum, and let $\tau(n)$ be the divisor function. Is there an asymptotic formula for $$\sum_{n\le x}\tau(n)c_q(n)$$ with error terms that do not depend on $q$?

In a paper of Soundararajan, equation (16c) states that for any Hecke eigenform $f$ of weight $k$, the symmetric square L-function at $1$ satisfies the bound $$(\log k)^{-2}\ll L(1, \mathrm{sym}^2f)\l …

It is a well known result that with $\tau(n) := \sum_{d|n} 1$ that $$\sum_{n\le X}\tau(n)\sim X\log X,\\ \sum_{n\le X}\tau(n)^2\sim C_1X\log^3X$$ for some $C_1 > 0$. In general, it is also known that …

Let $v(\beta) := \sum_{n\le X} e(n\beta)$ where $e(\alpha) := e^{2\pi i \alpha}$. It is not hard to show that $$\log X\ll \int_0^1 |v(\beta)| d\beta\ll \log X$$ and by considering the underlying Diop …

One version of Weyl's inequality states that for any $\alpha\in\mathbb{R}$ and $(a, q) = 1$ such that $|\alpha - a/q|\le 1/q^2$, we have that $$\sum_{n\le X} e(n^k\alpha)\ll X^{1 + \varepsilon}(q^{-1 …

There is a paper by Rizzo in which tables are given that specify the local root numbers of an elliptic curve based on $(a, b, c)$ where $(a, b,c)$ are non negative and minimal so that $a\equiv c_4\pm …

A well known useful inequality of Gallagher states (in one form) that for any sequence $a:\mathbb N\to\mathbb C$, we have that $$\int_{|\theta|\le\delta} \bigg|\sum_n a(n)e(n\theta)\bigg|^2d\theta\ll …

Given some $X, Y\ge 1$ and some $d\le Y$ not a perfect square, is it possible to bound $$\sum_{p\le X}\left(\frac{d}{p}\right)?$$ As long as $Y$ is not too large compared to $X$, I would expect that …

Does anyone know where I might be able to locate on the internet the following paper of Jutila?: M. Jutila, Mean value estimates for exponential sums. Number Theory, Ulm 1987, 120-136.