All Questions
37 questions with no upvoted or accepted answers
9
votes
0
answers
232
views
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;...
- 2,235
8
votes
0
answers
418
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\...
- 91
8
votes
0
answers
398
views
$L^1$ norm of Fourier transform of subset sums
Let $n_1,\dots,n_k$ be a set of $k$ natural numbers less than $N$, with $k = (1- \delta) \log_2 N$ for $\delta$ relatively small. Let $e(x) = e^{ 2\pi i x}$, as usual.
Assume that $$\int_0^1\prod_{j=1}...
- 148k
5
votes
0
answers
104
views
Exponential sums with monomials with divisor-function coefficients
In their paper "Exponential Sums with Monomials," Fouvry and Iwaniec study exponential sums roughly of the form
$$
\sum_{m_1 \sim M_1} \cdots \sum_{m_r \sim M_r} c_1(m_1) \cdots c_r(m_r) e\...
- 1,990
5
votes
0
answers
302
views
Exponential sums with prime power modulus
I am looking for an analogue of the following result of Fouvry and Katz for prime power modulus ("A general stratification theorem for exponential sums, and applications", J. reine angew. Math. 540 (...
- 26.9k
5
votes
0
answers
124
views
Linear exponential sum with gcd
The sum $$\sum _{d,d'\leq D}\sum _{h,h'=1}^q(h,q)e\left (\frac {dh+d'h'}{q}\right )$$ is easily seen to be $$\ll q^{2+\epsilon }+D^2.$$ Indeed with a standard estimate for a linear exponential sum it ...
- 143
4
votes
0
answers
513
views
Deriving inequality (8.9) from (8.8), in Iwaniec–Kowalski “Analytic Number Theory”
I am working through the problem presented in Chapter 8 of Iwaniec and Kowalski’s Analytic Number Theory (specifically inequalities (8.8) and (8.9)) and I am struggling with the transition between ...
- 111
4
votes
0
answers
78
views
Repeated values of a monomial
Let $H,M\geq 1$ and let $h_0$ and $m_0$ be fixed integers with $(h_0,m_0)\in [H,2H]\times[M,2M]$. Let $\alpha$ be a positive real number. I'm trying to find an upper found for the number of integer ...
- 1,990
4
votes
0
answers
168
views
Question about exponent pairs
In some of my recent research efforts, I've been applying a lot of estimates for exponential sums involving exponent pairs. Two seemingly simple questions have arisen from these calculations, and I ...
- 1,990
4
votes
0
answers
220
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 ...
- 1,381
4
votes
0
answers
169
views
Smoothed Weyl sum inequality
One version of Weyl's inequality states that for any $\alpha\in\mathbb{R}$ and $(a, q) = 1$ such that $|\alpha - a/q|\le 1/q^2$, we have that
$$\sum_{n\le X} e(n^k\alpha)\ll X^{1 + \varepsilon}(q^{-1}...
- 1,890
4
votes
0
answers
562
views
Best known bounds on certain exponential sums
What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...
- 1,890
3
votes
0
answers
179
views
Generalizing an estimate of Jutila
I'm working on a problem right now in which I need an upper bound for an exponential sum of the form
$$
\tag{1}
\sum_{N < n \leq 2N} \tau_3(n) e(f(n)),
$$
where $\tau_3(n) = \sum_{d_1d_2d_3=n} 1$ ...
- 1,990
3
votes
0
answers
127
views
A good way to bound the following exponential sum over $\mathbb{Z}/q\mathbb{Z}$ involving linear forms?
Let $q \in \mathbb{N}$. I am interested in getting an upper bound for the sum
$$
\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)...
- 3,625
3
votes
0
answers
206
views
Cancellation in this exponential sum?
I would like to know whether it is possible to obtain cancellation in the sum
$$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$
where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.
- 11.3k
3
votes
0
answers
340
views
On discrepancy of integer sequences related to Erdos-Turan-Koksma
Assume we have a sequence $a_1,\dots,a_k\in\Bbb N$ with each $a_i\approx n$ where $n$ is some integer.
Suppose there exists an $\eta\in(0,1)$ such that for every $d_1,\dots,d_k\in\Bbb Z$ with $$...
user94040
3
votes
0
answers
263
views
Number of solutions to $x_1x_2=x_3x_4\bmod n$
In https://www.math.ksu.edu/~cochrane/research/xyuvmodp.pdf it is shown $x_1x_2=x_3x_4\bmod p$ where $p$ is a prime has $\frac{|\mathcal B|}p+O(\sqrt{|\mathcal B|}\log^2p)$ solutions $(x_1,x_2,x_3,...
- 13.9k
2
votes
0
answers
191
views
The exponential sum over primes on average
In https://academic.oup.com/blms/article-abstract/20/2/121/266256?redirectedFrom=fulltext Vaughan shows the following bounds for the $L^1$-mean of the exponential sum over primes $$\sqrt x\ll \int _0^...
- 1,381
2
votes
0
answers
143
views
The exponential sum of $\omega (n)$
Let $\omega (n)$ be the number of (distinct) prime divisors of $n$ $$\omega (n)=\sum _{p|n}1$$ and let $S(a/q)$ be its exponential sum $$\sum _{n\leq x}\omega (n)e(na/q).$$
Question 1: Can anyone give ...
- 1,381
2
votes
0
answers
128
views
Distribution of square roots (mod m) on small intervals (with respect to m)
Fix a large positive integer $m$. Let $A$ be small positive number typically $\sim m^{1/3}$. Suppose $S(A, m)$ be set of solutions (normalized by dividing by $m$) to the quadratic congruences $x^2 = a ...
- 577
2
votes
0
answers
154
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'...
user147650
2
votes
0
answers
159
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.
- 1,890
2
votes
0
answers
249
views
An exponential sum like the Kloosterman sums
I encounter a tricky sum like the Kloosterman sum
$${\sum_{x \mod P}}^\ast e\left(\frac{ax+\overline{x}}{P}+\frac{lx^2}{P^2}\right),$$
where $l$ is a positive integer co-prime with $P$ and here $P$ ...
- 353
2
votes
0
answers
330
views
On exponential sum weighted with von-Mangoldt function
Suppose we have $\alpha \in \mathbb{R}$ such that $|\alpha - a/q| < 1/q^2$,
where $(a,q)=1$. Then we know that the exponential sum
$$
S(\alpha) = \sum_{1 \leq n \leq X} \Lambda(n) e(n \alpha)
$$
...
- 3,625
2
votes
0
answers
227
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 ...
- 12.8k
1
vote
0
answers
108
views
Manyfold iterated exponential sum with growing conductor
Let $\varphi(x_1,\dots, x_k)$ be some smooth function with partial derivatives of magnitude $\asymp 1$ for $x_i\asymp 1$. For concreteness, as it doesn't appear to add much extra structure beyond the ...
- 1,890
1
vote
0
answers
63
views
Optimal exponents in upper bound for 4-dimensional exponential sum
A classical result of Fouvry and Iwaniec states that if $\alpha_1,\ldots, \alpha_4$ are nonzero, $M_1,\ldots, M_4 \geq 1$, $X > 0$, and $|\varphi_{m_1,m_2}|,|\psi_{m_3,m_4}|\leq 1$ are complex ...
- 1,990
1
vote
0
answers
142
views
Partial exponential sums over lattice points of lattice cones
Consider the usual lattice $M:=\mathbb{Z}^2\subseteq\mathbb{R}^2$, and
let $v_1,v_2\in\mathbb{Z}^2\subseteq\mathbb{R}^2$ be two non-zero lattice points which are $\mathbb{Z}$-linearly independent. ...
- 7,609
1
vote
0
answers
94
views
Large sieve inequality-like sum without the square
Let $S(\alpha) = \sum_{n\leq N} w(n) e^{2\pi i \alpha n}$ for some function $w$ defined on $\mathbb{R}$. Suppose $\alpha_1, \ldots, \alpha_R$ are real numbers that are $\delta$-spaced modulo $1$, for ...
- 579
1
vote
0
answers
243
views
Sums of Kloosterman sums
Let
\[ S_{n,m}(q)=\sum_{a=1\atop {(a,q)=1}}^qe\left (\frac {an+\overline am}{q}\right )\]
be Kloosterman's sum and $\alpha _n,\beta _m$ be complex numbe of modulus $\leq 1$. For $Q,N,M>0$ what is ...
- 1,381
1
vote
0
answers
150
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
$$
\mathcal{I}_{p,N}=\int_0^1 \Big(\sum_{j=1}^N e^{2\pi i j \xi}+\sum_{j=1}^N e^{-2\pi i j \xi}...
- 421
1
vote
0
answers
56
views
Discrepancy bound of integer tensor product sequence?
Here discrepancy is from $(2.4)$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The (extreme) discrepancy $D_N(P) = D_N(x_l,\dots,X_N)$ of
the point set $P$ of $N$ points in $...
- 13.9k
1
vote
0
answers
118
views
Exponential sum over polynomial values
Let $f(x)=a_{k}x^{k}+\dots+a_{1}x+a_{0}\in \mathbb{R}[x]$ with $a_{k}>0$ and $N\in\mathbb{N}$ sufficiently large. I would like to know an estimate of the following sum:
$$\sum_{N\leq n\leq 2N}\exp(...
1
vote
0
answers
163
views
Is this averaged exponential sum over primes small infinitely often?
Do there exist infinitely many positive integers $N$ such that
$$\sum_{\substack{N/2 \leq q \leq N \\ a/q \notin \mathbb{Z}}} \left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \right|\leq |a|^{o(1)} N^...
- 217
1
vote
0
answers
165
views
Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$
Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let
$$
\mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} .
$$
We ...
- 3,625
0
votes
1
answer
347
views
Weyl sums in the arithmetic progressions
For any $\alpha \in \mathbb{R}$ which has the Diophantine
Approximation that $$\alpha=\frac{l}{q}+\frac{\theta}{q^2},\quad (l,q)=1, \quad|\theta|\le 1.$$ It is known that
$$\sum_{m\le M} \min \left(N,...
- 563
0
votes
0
answers
169
views
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 ...
- 3,625