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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Complexity of small near-reciprocals at $\frac34+\epsilon$ exponent - square free smooth number case

Let $q$ be large composite square free $O(\operatorname{polylog}(T))$-smooth number in $[T,2T]$ where $T$ is a parameter. According to the paper https://arxiv.org/abs/1103.2879 we can have many ...
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Complexity of small near-reciprocals at $\frac34+\epsilon$ exponent - prime power number case

Let $q=p^r$ be large prime power of prime $p$ with $q\in[T,2T]$ where $T$ is a parameter. According to the paper https://arxiv.org/abs/1103.2879 we can have many integer pairs $a,b$ satisfying $ab\...
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Complexity of small near-reciprocals at $\frac34+\epsilon$ exponent - prime number case

Let $p$ be large prime in $[T,2T]$ where $T$ is a parameter. According to the paper https://arxiv.org/abs/1103.2879 we can have many integer pairs $a,b$ satisfying $ab\equiv c\bmod p$ such that $|c|$ ...
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Self-referencing recurrence relation with exponential

I have the self-referencing recurrence relation $$ d(0) = 0 $$ $$ d(1) = a $$ $$ d(n+1) = d(n) + a*e^{-(\frac{d(n)-n*c}{b})^4} $$ Written as a sum: $$ d(N) = a*\sum_{n=0}^{N}e^{-(\frac{d(n)-n*c}{b})^...
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Bound for sum of multiplicative character calculated over multivariate polynomial

Let $f \in \mathbb{F}_q[x_1, \dots, x_k]$ be a polynomial with $\deg f = n$, and let $\chi$ be a multiplicative character over $\mathbb{F}_q$. Is there any known bound, possibly with conditions about $...
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Moments of an exponential sum

Let $p$ and $N$ be large natural numbers. I would like to get a possibly sharp asympotic approximation of $$ \mathcal{I}_{p,N}=\int_0^1 \Big(\sum_{j=1}^N e^{2\pi i j \xi}+\sum_{j=1}^N e^{-2\pi i j \xi}...
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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 $0$ ...
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A conjecture relating an integral and a sum, the floor function and squares

I've found through evidence and have conjectured on a math publication that: $$\Big\lfloor\int_1^\infty (k^{1/(k^{1+1/\sqrt{x}})} - 1)dk\Big\rfloor = \Big\lfloor\sum_{k=1}^{\infty}k^{1/(k^{1+1/\sqrt{x}...
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Why are exponential sums so bad at solving this very easy problem?

Exponential sums are a powerful tool in additive combinatorics and number theory. In my understanding, when it comes to estimate the cardinality of a certain set, exponential sums are (essentially) ...
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Sum of Kloosterman sums with oscillating factor

Denote by $S(c;n,m)$ Kloosterman's sum. Take $X>0$ and take $n,m\in \mathbb Z$ smaller than a small power of $X$ in modulus. It is known that essentially \[ \sum _{c\sim X}\frac {S(c;n,m)}{c}\ll ...
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Bounds on exponential and character sums of ratio of linear recurrences

Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\chi$ be a non-trivial additive character of $\mathbb{F}_q$, and let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$. Also,...
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growth rate of quadratic exponential sums with monomial coefficients

What is the growth rate of $$S_d(M)=\sum_{n=1}^M n^d e\left(\frac{n^2}{2M}\right)$$ where $M$ is an even integer. My numerical experiments show that $$\frac{S_d(M)}{M^{d+\frac{1}{2}}}\approx e^{\frac{\...
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What does this exponential sum evaluate to?

We have the following sum $$S=\sum_{\substack{0<a'\leq k'\\(a',k')=1\\a'\equiv b \bmod q}}e(hl'a'/k').$$ Here, $e(x):=\exp(2\pi i x)$, $h,k',q,a'$ are all natural numbers. We do know that $\gcd(h,l'...
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Reference for a paper of Jutila

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.
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Exponential Series with a sequence [closed]

For a convergent sequence $(a_n)_n \rightarrow a$ consider the exponential series \begin{equation*} \exp_{(a_n)_n}(-x) := \sum_{n=0}^{\infty} \frac{(-x)^n a_n}{n!}. \end{equation*} Can there be ...
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Which unimodular lattices $L\subset \mathbb R^2$ minimize $f_t(L):=\sum_{ v\in L} e^{-t \|v\|_2}$? (for parameters $t>0$)

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Consider the lattices in $\SL(2,\mathbb R)(\mathbb Z^2)$ up to rotation. The space of such lattices can be identified with the modular surface $\...
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5 votes
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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 ...
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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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12 votes
1 answer
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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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6 votes
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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 ...
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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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Polynomial composition utilizing polynomials in two different finite fields

At every $n\in\mathbb N$ (all polynomials are of degree $O(1)$) is there $g_{3,1}^{(n)},\dots,g_{3,k}^{(n)}\in\mathbb F_3[x_1,\dots,x_n]$ at $k=\mathsf{poly}(n)$ and $g_2^{(n)}\in\mathbb F_2[x_1,\dots,...
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Can the following sum be counted or expressed in terms of special functions?

Let us define this sum as a function of $z \in \mathbb{C}$ with some positive parameter $a$ $$ f(z; a) = \sum\limits_{n = 0}^{\infty}\frac{|z|^{2n}}{n!}e^{-ian^2}. $$ Probably, it can be expressed (or ...
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2 answers
247 views

Closed form for $\sum_{i=1}^n{a^{i^2}}$

Let $a$ be an element of some ring or field, possibly finite. Is there closed form for $\sum_{i=1}^n{a^{i^2}}$? sage and wolframalpha couldn't solve it. If $a$ is primitive n-th root of unity this is ...
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What is the bound for $L_{\infty}$ norm for positive part of exponential sum?

A famous conjecture posed by Littlewood and solved by O. McGehee, L. Pigno, and B. Smith(in their article, Hardy's inequality and the $L_1$-norm of exponential sums) and S. V. Konyagin independently ...
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Is this limit zero?

Define $e(\theta)=e^{2\pi i\theta}, \theta\in [0,1]$, $P_n=\{p_1,p_2,...,p_n\}$ are the first $n$ primes, $\|f\|_1=\int_{[0,1]}|f(\theta)|d\theta$. Problem 1. is it true for all fixed $m\in \mathbb{N^...
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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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8 votes
0 answers
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Explicit bound for sum of Kloosterman sums

What are the best fully explicit upper bounds one can give for the sum $$\left\lvert \sum_{n=N}^{\infty} \frac{S(a,b;n)}{n} \,I_1\!\left(\frac{4 \pi \sqrt{|ab|}}{n}\right) \right\lvert$$ where $S(a,b;...
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3 votes
1 answer
139 views

Exponential sums over rings

I'm trying to evaluate an exponential sum of the form: \begin{equation} \sum_{c\in Z_q}\chi(f(c)) \end{equation} For polynomial $f(x)=a_2x^2+a_1x+a_0$ (with $a_2\ne 0$). If $q$ is prime, then this is ...
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4 votes
1 answer
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How different can the bias of two polynomials be?

I'm trying to figure out how to approach the following question: Let $g,h$ be polynomials over $\mathbb{Z}_p$ (for prime $p$) with $n>1$ variables. Denote by $bias(g)=|\sum_{x\in \mathbb{Z}_p^n}e^{...
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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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Find conditions for the following running average to be monotonically decreasing

Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [px_t^2 - (p+q)x_t]$ where $x_t = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $...
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Prove that the following running average is monotonically decreasing

Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [x^2+(p-q)x]$ where $x = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $n$. $0 &...
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7 votes
1 answer
450 views

Riemann hypothesis for exponential sum

Recently I've heard about the Riemann hypothesis for one-variable exponential sums, which states as For a polynomial $f\in\mathbb{F}_{p^k}[x]$ of degree $d$ and a character $\chi$ of $(\mathbb{F}_{p^...
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8 votes
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375 views

Equidistribution of $\{\sqrt{p}: p \text{ primes }\}$ modulo 1

I am trying to show $\{\sqrt{p}: p \text{ primes }\}$ is equidistributed modulo 1. Using Weyl's criterion, it is sufficient to show for each nonzero integer $k$, \begin{equation} \sum_{n \leq x}e(k\...
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4 votes
1 answer
207 views

Mean square estimate for the Kloosterman sums

For $m,n\in \mathbb{N}$, denote the Kloosterman sum $$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$ denotes the multiplicative inverse of $a\bmod c$. Does ...
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2 votes
1 answer
180 views

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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7 votes
1 answer
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Upper bound for an exponential sum involving characters of a finite field

Let $q = p^n $ be a prime power, $\alpha\in\mathbb{F}_{q} $ a primitive element of the finite field $\mathbb{F}_q$ and denote by $\chi $ a non-trivial additive character of $\mathbb{F}_{q} $. Set $\...
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Number of integer solutions to quadratic polynomial with integer coefficients

It is known from for example Representations of Integers as Sums of Squares by Grosswald, E. that $$|\{(n_1,n_2,\ldots,n_k)\in\mathbb{Z}^k: \ n_1^2+n_2^2+\cdots+n_k^2=N\}|\leq C_\varepsilon N^{\frac{...
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4 votes
1 answer
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Estimating certain short Kloosterman sums

Recall that for the classical Kloosterman sum $$ K(a,b,p^t):= \sum_{x \in (\mathbb{Z}/ p^t \mathbb{Z})^* } \psi \left(\frac{ax+bx^{-1}}{p^t} \right),$$ where $\psi(x)=e^{2\pi ix}$, $a,b,t$ are natural ...
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2 votes
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Converse of Gallagher identity

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 \...
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18 votes
2 answers
2k views

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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1 vote
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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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1 vote
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191 views

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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1 vote
0 answers
74 views

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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11 votes
2 answers
487 views

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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3 votes
0 answers
122 views

A good way to bound the following exponential sum over $\mathbb{Z}/q\mathbb{Z}$ involving linear forms?

Let $q \in \mathbb{N}$. I am interested in getting an upper bound for the sum $$ \sum_{(a_1, a_2, a_3, q) = 1} \sum_{\mathbf{h} \in (\mathbb{Z}/q\mathbb{Z})^n }e( \frac{a_1}{q}\ell_1(h_1, \ldots, h_n)...
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