# 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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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 $ab\... 0 votes 0 answers 89 views ### Complexity of small near-reciprocals at$\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|$... 1 vote 0 answers 100 views ### 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: $$d(N) = a*\sum_{n=0}^{N}e^{-(\frac{d(n)-n*c}{b})^... 4 votes 1 answer 112 views ### Bound for sum of multiplicative character calculated over multivariate polynomial Let f \in \mathbb{F}_q[x_1, \dots, x_k] be a polynomial with \deg f = n, and let \chi be a multiplicative character over \mathbb{F}_q. Is there any known bound, possibly with conditions about ... 1 vote 0 answers 109 views ### Moments of an exponential sum Let p and N be large natural numbers. I would like to get a possibly sharp asympotic approximation of$$ \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}... 1 vote 1 answer 105 views ### 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 ... 0 votes 1 answer 294 views ### 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$0$... 2 votes 1 answer 203 views ### 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}... 8 votes 1 answer 368 views ### 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) ... 4 votes 0 answers 159 views ### 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 ... 6 votes 0 answers 89 views ### 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,... 3 votes 0 answers 68 views ### 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$$\frac{S_d(M)}{M^{d+\frac{1}{2}}}\approx e^{\frac{\... 2 votes 0 answers 144 views ### What does this exponential sum evaluate to? We have the following sum $$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'... 142 views

### 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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### 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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### 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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### 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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### 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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### 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 ...
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 \...