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 ...

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2
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1answer
66 views

Elaboration of a certain section of a paper by Thanigasalam

In section 11 of this paper by Thanigasalam, it says "... we get $G(10)\le 105$, and this implies that $H(10) \le 107$". However, it is very unclear how this follows. Why is it the case that $G(10)\le ...
2
votes
0answers
100 views

Twisting by a multiplicative Character in Katz, Perversity and Exponential sums

Let $C(x_1,\ldots,x_n)$ be a nonsigular cubic form with integral coefficients. In his Proof that $C$ fulfills the Hasse-Principle, if $n\geq 9$, Hooley used the following estimate that was provided ...
5
votes
1answer
266 views

Sums of random variables mod p

Let $\varepsilon_1, \ldots, \varepsilon_n$ be independent random variables taking values $0,1$ each with probability $1/2$. It is well known that $R_n=\varepsilon_1+ \cdots+ \varepsilon_n$ modulo a ...
4
votes
0answers
370 views

Best known bounds on certain exponential sums

What are the best bounds currently known for the following exponential sum: $$\sum_{x < p \le 2x} e(\alpha p^k)$$ for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...
9
votes
2answers
560 views

Iwaniec-Kowalski Exponential Sum for Quadratic Function

I am reading about 'Exponential Sums' in the book 'Analytic Number Theory' by Iwaniec and Kowalski. On page 199 they mention the bound: $$|S_f(N)|^2 \le N +2N^2q^{-1}+4(N+q)\log q \tag{1}$$ where, ...
9
votes
1answer
297 views

$L^1$ norm of exponential sum of $n^2 x$

What is the asymptotic order of $$ \int_0^1 \left| \sum_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx $$ as $N \to \infty$. This should be known, but I cannot find it in the literature.
0
votes
1answer
70 views

What is the maximal number of solutions of $\sum_{i = 1}^n 1/a_i^x - \sum_{i = 1}^m 1/b_i^x = 0$?

What is the maximal number of solutions of the following equation? $\sum_{i = 1}^n 1/a_i^x - \sum_{i = 1}^m 1/b_i^x = 0$ where $x$ is the unknown and $n, m$, $a_i$'s, $b_i$'s are constant. It ...
1
vote
1answer
114 views

Uniform convergence of infinite sum with Dirichlet characters

I would like to prove uniform convergence of function series like : $$\sum\limits_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in ...
1
vote
3answers
146 views

Decidability of sum of powers exponential diophantine equation

I want to ask about decidability of exponential Diophantine equation: $z_12^{\eta_1} + \ldots + z_n2^{\eta_n} = z$, where $z_i,\eta_i,z \in \mathbb{Z}$, and $\eta_i$ - are variables. Can we find ...
0
votes
0answers
72 views

separating parameters in generalized quadratic Gauss sum

The normalized generalized quadratic Gauss sum is defined by $$ G(a,b,c)=\frac{1}{c}\sum_{n=1}^ce\left(\frac{an^2+bn}{c}\right) $$ where $e(x)=\exp(2\pi ix)$. Under what conditions on $c$ can we ...
3
votes
0answers
233 views

Solving a doubly exponential generating function

I am analyzing the average time complexity of some algorithm on some probabilistic model, and I've come to a doubly exponential sequence for which I cannot find corresponding generating function. I ...
2
votes
1answer
135 views

Bounds on imaginary parts of partial Kloosterman sums?

For a prime $p$ and integers $a,m$, $0<a,m<p$ define the (partial Kloosterman) sum $$ S_p(a,m) = \sum_{0<k<m} \exp\left(\frac{2\pi\mathrm{i}}{p}(a x + x^{-1})\right), $$ where $x^{-1}$ is ...
2
votes
2answers
217 views

Bound on exponential sum with weights

Let $e(z)$ denote $e^{2 \pi i z}$ and let $f(z)$ a smooth real function. I know one can bound sums of the form $$ \sum_{x \leq X} e(f(x)) $$ via for example Van der Corputs's result, provided we make ...
0
votes
0answers
53 views

Definition of Degenerate hyper-Kloosterman Sum

If $(a,q)=1$, we all know the usual hyper-Kloosterman sum $$Kl_n(a,q)= \sum_{x_1...x_n\equiv a \mod q} e(\frac{x_1+\dots+x_n}{q}).$$ I wonder what would be the correct definition when $(a,q)\neq 1$. ...
8
votes
2answers
526 views

Incomplete Kloosterman sum

I am interested in an upper bound on the following incomplete Kloosterman sum $$ \sum_{\substack{x=1 \\ x+_{_{\bf Z}}x^{-1}>p}}^{p-1}e\left(\frac{x+x^{-1}}{p}\right).$$ Using the Weil's bound it ...
2
votes
0answers
166 views

Kloosterman-like sum with inverse to different moduli

In some recent work, the following strange-looking exponential sum arose: $$ \sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg). $$ Here $e(x) = e^{2\pi i x}$ as ...
1
vote
1answer
464 views

stationary phase method in analytic number theory

I hope someone can tell me something about the error term in the formula calculating the oscillatory integral like $\int_a^b g(x)e(f(x))d x$. Specially, the exact formula on page 114 of M. Huxley's ...
1
vote
1answer
580 views

Exponential sums

I would like to estimate the following sum $\sum_{N <n \leq 2N}e(vn^{l})$, $l \geq 1$ constant(not integer) and $v$ is a parameter(integer) that doesn't grow too fast(a small power of N). The ...
21
votes
1answer
1k views

Is there a cheap proof of power savings for exponential sums over finite fields?

Let $p$ be a large prime, and let $f(x) = P(x)/Q(x)$ be a non-constant rational function over ${\Bbb F}_p$ of bounded degree. From the Weil conjectures for curves, we have a bound of the form $$ ...
3
votes
3answers
643 views

iteratively (approximately) solving a sum of exponentials

I would iteratively have to solve the following equation at iteration $n$: $C = \sum_{1 \leq i \leq n}{e^{\frac{x_i}{T}}x_i}$ for $T$. Each iteration $i$ an unknown $x_i$ will be observed and $C$ is ...
14
votes
1answer
810 views

On the $L^1$-norm of certain exponential sums.

I am stuck with an elementary-looking problem, which does not belong to my usual field of research so I eventually decided to ask it on MO. Let $S$ be a finite set of integers. For $P$ a subset of ...
7
votes
1answer
470 views

Lower bound for exponential sums.

Let $D$ be a subset of $\mathbb Z/n \mathbb Z$ containing $0$. For $m$ an integer, set $$\alpha(m,D)=\sum_{d \in D} e\left (\frac{m d }{n}\right ),$$ where as usual $e(x) = e^{2 i \pi x}$ This is an ...
5
votes
1answer
263 views

Where can I read about exponential sums corresponding to Jones Polynomial?

I remember reading that a number theoretic analogue of Witten's path integral formula for the Jones polynomial: $$\text{Jones}_K(e^{2\pi i/(k+2)})=\int_{\text{$SU(2)$ connections on $\mathbb ...