# 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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### Integral over an exponential sum with squares

How should I estimate the following integral $$I = \int_0^1 \left( \sum_{n=0}^{p-1} e(n^2t) \right)^2 dt$$ where $p$ is a prime? Here is the method I followed: \begin{align*} I & = \int_0^1 \...
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### A finite alternating sum

We have stumbled upon the following finite alternating sum, which we have trouble analyzing. The sum is: $$S_n = \sum_{j=0}^n \frac{ (-1)^j e^{-j} }{j!} (n-j)^j$$ We have observed numerically that ...
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### An integral involving many exponential terms with quadratic exponents in the denominator

Given $k$ points $\{p_1,\cdots, p_k\}$ in $\mathbb{R}^n$ and positive constants $r_1, ..., r_k$ and another positive constant $\alpha>0$. Is there a way to compute/approximate the following (...
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### Way to express a number in its most compact sum of powers

Given a non-negative number n, what is the best approach to find the most compact representation for n in terms of sums of powers, such that the bases and the exponents can't surpass a given value (...
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### Formula for exponential integral over a cone

While reading 'Computing the Volume, Counting Integral points, and Exponential Sums' by A. Barvinok (1993), I came across the following: "Moreover, let $K$ be the conic hull of linearly independent ...
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### A question regarding Bourgain's paper on $\Lambda(p)$-subsets

I'm trying to understand Bourgain's proof of Proposition 1.10 on page 304-307 in On $\Lambda(p)$-subsets of squares which states Given $p>4$, we have the estimate \begin{align} \left\|\sum_{n=...
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### Coefficients $U_m(n,k)$ in the identity $n^{2m+1}=\sum\limits_{0\leq k \leq m}(-1)^{m-k}U_m(n,k)\cdot n^k$

Review the main result of mathoverflow.net/questions/297900, that is the identity \begin{equation}\label{f1} n^{2m+1}=\sum\limits_{1\leq k \leq n}\sum\limits_{j\geq0}A_{m,j}k^j(n-k)^j, \end{equation} ...
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### Explicit numbers with square root cancellation in Weyl's exponential sum

I'm interested in examples of real numbers $\alpha$ where we have $$\left| \sum_{n=1}^N \mathrm e(\alpha n) \right| \ll N^{1/2}$$ or perhaps with the weaker estimate with the right side replaced ...
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### Summing the infinite series $\sum_{k=0}^{\infty} \frac{x^k}{(k!)^2}$ [closed]

Is there a closed form sum of $\sum_{k=0}^{\infty} \frac{x^k}{(k!)^2}$ It is trivial to show that it is less than $e^x$ but is there a tighter bound? Thanks
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### Prerequisites to read Katz's Gauss Sums, Kloosterman Sums, and Monodromy Groups

What is the minimal background one must have to read Katz's Gauss Sums, Kloosterman Sums, and Monodromy Groups?
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### An exponential sum estimate on small intervals

Let $1<r<2$ be a real number. Let $4<p\le 6$. Consider the exponential sum estimate $$\int_0^{2\pi}\int_0^{N^{r-2}} \left|\sum_{n=1}^N e^{inx+in^2 y}\right|^p \, dy \, dx$$ Notice that the $y$...
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### Upper bound for $\sum_{p\leq x} a(p)p^{it}$

The Vinogradov-Korobov zero-free region for the Riemann Zeta-function would give us an upper bound for $\sum_{p\leq x} p^{it}$. Now my question is, what would be the corresponding upper bound for ...
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### Bounding exponential sum of the form $\sum_{\mathbf{x} \in (\mathbb{Z}/q \mathbb{Z})^n } \chi_1(x_1)\cdots \chi_n (x_n) e(a F(\mathbf{x})/q)$

I have encountered the following exponential sum and I would like to obtain a non-trivial upper bound for it. I am not quite sure where to start, and I would greatly appreciate any suggestions on how ...
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### Hypotheses for exponent pairs

The theory of exponent pairs provides bounds for $$\sum_{N<n<2N} e(f(n)),$$ where f behaves like a monomial. Precise formulations of this are in Graham and Kolesnik (GK) which seems to be what ...
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### optimal estimate for generalized Kloosterman sum

Let $p$ be an odd prime. Denote $e(x):=e^{2\pi i\frac{x}{p}}$. Let $n\ge 2$ be an integer. Consider the exponential sum $$S(f,g)=\sum_{g(x_1,\dots,x_n)=0, x_i\in\mathbb{F}_p}e(f(x_1,\dots,x_n)),$$ ...
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Let $p$ be a prime and consider the field $\mathbb{F}_p$. Fix $f\in\mathbb{F}_p[X]$ a polynomial of degree $d\ge 2$. Define $$K(x,y)=\frac{1}{\sqrt{p}}\sum_{z\in\mathbb{F}_p}e_p(xz+yf(z)),$$ where $... 1answer 513 views ### exponential sum over variety I am wondering where to find a good reference for bounds of the type $$\sum_{x\in V(\mathbb{F}_p)} \chi(g(x))\psi(f(x))$$ where$V$is a variety,$\chi$is a multiplicative character over$\mathbb{...
Let $p$ be a prime, let $\zeta_p=e^{2\pi i/p}$, let $g\in{\bf F}_p$ be a non-square and let $\chi:{\bf F}_p^*\rightarrow{\bf C}^*$ be a non-trivial character. Then the complex numbers  \chi(n)\...
### How many integers between $\left[2^{2^k}, 2^{2^{k+1}}\right]$? [closed]
Suppose $k$ and $n$ are natural numbers such that $2^{2^k} \lt n \lt 2^{2^{k+1}}$. I am curious how many integers are there in the interval $\left[2^{2^k}, 2^{2^{k+1}}\right]$ in terms of $n$. I need ...