# Questions tagged [exponential-sums]

The method of exponential sums is one of a few general methods enabling us to solve a wide range of miscellaneous problems from the theory of numbers and its applications. The strongest results have been obtained with the aid of this method. Therefore knowledge of the fundamentals of theory of exponential sums is necessary for studying modern number theory.

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### Bounds on quadratic character sums

I asked this question on Mathematics stack exchange but didn't get a response, so I ask here too. Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free ...
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### Manyfold iterated exponential sum with growing conductor

Let $\varphi(x_1,\dots, x_k)$ be some smooth function with partial derivatives of magnitude $\asymp 1$ for $x_i\asymp 1$. For concreteness, as it doesn't appear to add much extra structure beyond the ...
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### Exponential sum involving floor function

Can one get cancellation in exponential sums such as, say, $$\sum_{n\sim N} e(\lfloor n^\theta\rfloor^\beta),$$ for fixed positive $\theta,\beta\not\in\mathbb Z$? When $\theta < 1$, it seems ...
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### The Wilton-type bounds involving half-integral weight cusp forms

There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following: Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be ...
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### Exponential sum with weight in bottom

I am interested in the exponential sum $$\sum_{n=1}^X \frac{e(c_1n^2+c_2 n)}{1-e(c_1n)}$$ where $c_2$ is irrational and $e(x)=e^{2\pi i x}$. I know if the denominator is not there, this is a Weyl sum ...
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### Number of solutions of $am \equiv bn \pmod{q}$

Let $q$ be a (large) prime. Let $N$ be a positive integer of size${}\approx \sqrt{q}$. Let $\mathcal{M}$ be an arbitrary subset of $\{1, \dots, q\},$ such that $\mathcal{M}$ has cardinality $N$. ...
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### Proof of Szegö asymptotic theorem

Consider the truncated exponential series $$P_N(z) = \sum_{n= 0}^N \frac{z^n}{n!}$$ The zeros of this series have been studied by Szëgo and others (see e.g. here). He established an asymptotic for the ...
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### Partial exponential sums over lattice points of lattice cones

Consider the usual lattice $M:=\mathbb{Z}^2\subseteq\mathbb{R}^2$, and let $v_1,v_2\in\mathbb{Z}^2\subseteq\mathbb{R}^2$ be two non-zero lattice points which are $\mathbb{Z}$-linearly independent. ...
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### Exponential sum vs. exponential integral via Poisson summation

When we want to estimate an exponential sum $$\sum_{M<m\le M'}e(f(m)) \quad\text{with}\quad 1\le M\le M'\le 2M \quad\text{and}\quad e(x):=\exp(2\pi ix)$$ where $e(x):=\exp(2\pi ix)$ and the phase ...
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### A double sum with complex numbers having stochastic variables

I am very confused by a sum I have been trying to solve analytically/ numerically for a long time. It comes from the idea of a physical problem where the observation is made that has a combined ...
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### A basic estimate of exponential sums

Demeter in his book "Fourier Restriction, Decoupling, and Applications" (P287) used the following estimate: \sup_{0\leq n\leq q}\bigg|\sum_{m=0}^n e^{2\pi i\frac{a}{q}m^2}\...
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### Mean value of the divisor function over Piatetski-Shapiro sequences

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 ...
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### Large sieve inequality-like sum without the square

Let $S(\alpha) = \sum_{n\leq N} w(n) e^{2\pi i \alpha n}$ for some function $w$ defined on $\mathbb{R}$. Suppose $\alpha_1, \ldots, \alpha_R$ are real numbers that are $\delta$-spaced modulo $1$, for ...
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For any $\alpha \in \mathbb{R}$ which has the Diophantine Approximation that $$\alpha=\frac{l}{q}+\frac{\theta}{q^2},\quad (l,q)=1, \quad|\theta|\le 1.$$ It is known that $$\sum_{m\le M} \min \left(N,... • 553 2 votes 1 answer 183 views ### On the upper-bound for a type of quintuple Kloosterman sums Sorry to disturb, dear experts here. I have a question involving the quintuple Kloosterman sum, and expect some hints to show the upper-bound. My question is, for any x,y,z,w,\delta \in \mathbb{Z} ... • 553 14 votes 1 answer 632 views ### The first case of the strong Littlewood conjecture Let A be a set of n integers and consider the quantity:$$\int_{0}^1 \left| \sum_{a \in A} e^{2\pi i a x} \right|dx. $$The (now solved) Littlewood conjecture is the claim that this quantity is ... • 12k 3 votes 1 answer 333 views ### Estimates for certain double-Kloosterman sums Sorry to disturb. I encounter a double-Kloosterman sum, which needs some help from the experts here. For any q\in \mathbb{N}^+, how can we estimate the type of sum$$ \sideset{_{}^{}}{^{\ast}_{}}\...
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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 ...
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### A question involving the three-dimensional Kloosterman sum

Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here. For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$,...
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