# 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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### Solving the inequality between a and b [closed]

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 ...
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### An asymptotic expansion of a infinite sum

I am interested in the asymptotic expansion in $t$($t>0$) when $t\to 0^+$ of the following series $$\sum_{k\ge 0}e^{-k^{2/n}t}$$ for integer $n>2$ (n=1 follows from Poisson summation formula ...
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### Growth rate of exponential sum of $S_j$

Let $X_j$, $1 \leq j \leq n$ be chosen i.i.d. uniform over $[0,2\pi)$. Denote $S_j \triangleq X_1 +X_2+\cdots+X_j$ and suppose that $c_j$, $1 \leq j \leq n$ are some constants such that $|c_j|=1$. I'...
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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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