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# 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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### 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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1 vote
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### Optimal exponents in upper bound for 4-dimensional exponential sum

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 ...
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### Why are Deligne-type exponential sum estimates so hard to use?

Let $F$ by a finite field, and $R(x_1,x_2,\ldots,x_n) := r_1(x_1,x_2,\ldots,x_n)/r_2(x_1,x_2,\ldots,x_n)$ a rational function in $n$ variables, frequently in analytic number theory or harmonic ...
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### 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 $x,y,z,w,\delta \in \mathbb{Z}$ ...
588 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 ...
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1 vote
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### Sums of Kloosterman sums

Let $S_{n,m}(q)=\sum_{a=1\atop {(a,q)=1}}^qe\left (\frac {an+\overline am}{q}\right )$ be Kloosterman's sum and $\alpha _n,\beta _m$ be complex numbe of modulus $\leq 1$. For $Q,N,M>0$ what is ...
1 vote
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1 vote
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### Distribution of quadratic polynomials mod $n$ and $n^2$

Suppose $n$ is odd, then both equations $x^2 = D \; mod \;n$ and $x^2 = D \; mod \;n^2$ have the same number of solutions for fixed $D$ coprime to $n$. What can be said about the relationship between ...
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### The exploration of the asymptotic behavior of a simple sum. $\sum_{k=1}^{\infty} (k^{1/k} - 1)$ [closed]

An informal investigation of a sum. Consider this sum: $$S =\sum_{k=2}^{\infty}(k^{1/k} -1)$$ Does this converge? How does it behave as it diverges, if it diverges? If $k$ equaled $1$ we would get $0$ ... 265 views

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### Upper bound an integral with exponential function

I am working on my research about approximation a function. I come up with the following integral. I run some simulations and saw that the integral would converge to zero as n goes to infinty. Here is ...
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### Generalizing an estimate of Jutila

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 $\tau_3(n) = \sum_{d_1d_2d_3=n} 1$ ...
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### Product of three or more independent sub-Gaussian varibles

A random variable $X$ is called subgaussian of order $\sigma^2$ if $\log E[exp\{\theta X\}]\leq \frac{1}{2}\theta^2\sigma^2$ for every $\theta\in\mathbb R$. Given a sequence of independent subgaussian ...
1 vote
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### An inequality between sum of exponential functions wrt dyadic index

I am reading a paper 'Periodic Nonlinear Schrodinger Equation and Invariant Measures' written by J.Bourgain. And I am wondering if I can have some help from this website. My question is an inequality ...
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### Deligne's theorem on exponential sums

I'm an analyst who needs to use Deligne's Theorem 8.4 in 1, but I feel lost in the maze of definitions, and I don't trust my geometric intuition here. Theorem 8.4: Let $Q$ be a polynomial in $n$ ...
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### Maximum number of positive roots is $3$

Let $$f(x) = a+b(x+p)^t+c(x+p)^t(x+q)^t+d(x+p)^t(x+q)^t(x+r)^t,$$ where $t>1$ is any positive real number, $p>q>r>0$ or $p<q<r$ are positive integers and $a,b,c,d$ are any ...
I run into this inequality $$(a + b)^{1 - \epsilon} \;a < b$$ where $a \in \mathbb{Z}^+$ and $\epsilon \in (0, 1)$. What value (w.r.t $a$ and $\epsilon$) should I set $b$ equal to such that this ...