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17 votes
1 answer
593 views

Smoothed exponential sums: bounds and sources?

Let $f:\mathbb{R}\to\mathbb{C}$ be differentiable $k$ times, with $f, f',\dotsc,f^{(k)}\in L^1$. Let $\alpha\in \mathbb{R}/\mathbb{Z}$, $\alpha\ne 0$. In "Every odd number..." (Math. Comp. 83, 2014), ...
H A Helfgott's user avatar
  • 20.2k
16 votes
1 answer
1k views

On (a generalization of) the Gauss Circle Problem

Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...
Fan Zheng's user avatar
  • 5,169
12 votes
2 answers
1k views

counting points on unit sphere mod p

Let $f(n)$ be the number of points on the unit sphere $x^2 + y^2 + z^2 = 1\; \pmod n$ with $x,y,z \in \mathbb{Z}/n\mathbb{Z}$ This is sequence A087784 in the Online Encyclopedia of Integer ...
john mangual's user avatar
  • 22.8k
11 votes
2 answers
2k 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, $...
r9m's user avatar
  • 810
11 votes
2 answers
1k 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 ...
Farzad Aryan's user avatar
11 votes
1 answer
1k 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 ...
Joël's user avatar
  • 26k
10 votes
1 answer
474 views

A basic estimate of exponential sums

Demeter in his book "Fourier Restriction, Decoupling, and Applications" (P287) used the following estimate: \begin{equation} \sup_{0\leq n\leq q}\bigg|\sum_{m=0}^n e^{2\pi i\frac{a}{q}m^2}\...
Dapao Zhang's user avatar
9 votes
1 answer
729 views

Sums of twisted products of Kloosterman Sums

For $m,n,c \in \mathbb{N}$, let $S(m,n;c)$ denote the Kloosterman sum $$ S(m,n;c) := \sum_{\substack{1 \leq a < c \\ \gcd(a,c) = 1}} e \left( \frac{ma + n\overline{a}}{c} \right) $$ where $e(n) = e^...
davidlowryduda's user avatar
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;...
Fredrik Johansson's user avatar
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\...
Kyle Yip's user avatar
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}...
Will Sawin's user avatar
  • 148k
7 votes
1 answer
518 views

Does there exist some irrational $x,\alpha$ so that this Weyl sum is $o(\sqrt N)$?

This is a less ambitious version of Is the Lebesgue measure of the $x$ so that this exponential sum is $o(\sqrt{N})$ positive? . Consider $$S_N(x):=\sum_{n=1}^N \exp\left(2\pi i\left(\frac12n^2x+\...
user479223's user avatar
  • 1,904
7 votes
2 answers
531 views

Conjecture about an exponential sum

Let $X \subset \mathbb{N}$ and say that $X$ is super-equidistributed if for all $\alpha \in \mathbb{R} \setminus \mathbb{Z}$ there exists $C(\alpha) > 0$ such that for all $N$ $$ \left| \sum_{x \in ...
Lambert A'Campo's user avatar
6 votes
1 answer
285 views

What is the mean value of a pair of Ramanujan Sums when summed over squares?

Does anyone know of the mean value of two Ramanujan Sums when summed over the square of integers? In my research on the Landau problem regarding nearly square primes, I have run into the mean value ...
John Washburn's user avatar
6 votes
1 answer
283 views

Is the Lebesgue measure of the $x$ so that this exponential sum is $o(\sqrt{N})$ positive?

Consider $$S_N:=\sum_{n=1}^N \exp\left(2\pi i\left(\frac12n^2x+\alpha n\right)\right)$$ where $\alpha$ is irrational. For certain $x$ (say integer) we can get that this is bounded for all $N$. I am ...
user479223's user avatar
  • 1,904
6 votes
1 answer
287 views

Number of solutions for the inequality with square roots

Let $M$ be some large real number and $\delta>0$. I would like to estimate the number of solutions for the inequality $$|\sqrt{n_1}+\sqrt{n_2}-\sqrt{n_3}-\sqrt{n_4}|<\delta\sqrt{M},$$ where $...
Alexander Kalmynin's user avatar
6 votes
1 answer
292 views

Prime number theorem via large sieve type sums

We know that the prime number theorem is equivalent to the statement $$ M(x)=\sum_{n\le x}\mu(n)=o(x). $$ By using Ramanujan sums, we can write $M(x)$ as $$ M(x)=\sum_{q\le x}\sum_{\substack{0\lt a\le ...
Itachi's user avatar
  • 178
6 votes
2 answers
426 views

Average of gcd of sum of two $k$th powers

I am interested bounding the following quantity. Given fixed $k \in \mathbb{N}$, $a,b \in \mathbb{Z}$, $\sigma \in [0,1)$, and intervals $I_1,I_2 \subset \mathbb{Z}$ can we establish the bound $$S = \...
Daniel Flores's user avatar
6 votes
1 answer
183 views

Mean value of the divisor function over Piatetski-Shapiro sequences

Let $c>1$, $c\not\in\mathbb{Z}$ and consider the sum $$ \sum_{n\leq x} \tau(\lfloor n^c \rfloor), $$ where $\tau(n)$ is the number of divisors of $n$. I'm almost certain I've seen an evaluation of ...
Joshua Stucky's user avatar
5 votes
1 answer
405 views

Exponential sum involving floor function

Can one get cancellation in exponential sums such as, say, $$ \sum_{n\sim N} e(\lfloor n^\theta\rfloor^\beta), $$ for fixed positive $\theta,\beta\not\in\mathbb Z$? When $\theta < 1$, it seems ...
Mayank Pandey's user avatar
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\...
Joshua Stucky's user avatar
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 (...
Stanley Yao Xiao's user avatar
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 ...
caws's user avatar
  • 143
4 votes
1 answer
282 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 ...
FeiHou's user avatar
  • 353
4 votes
1 answer
776 views

An exponential sum over squares

I have the following exponential sum: $\sum _{M<n\leq N}e\left (x/n^2\right )=\sum f(n),$ say, where $M$ and $N$ are something like $x^{1/4}$ and $x^{1/2}$. My question is basically, how do I ...
tomos's user avatar
  • 1,381
4 votes
1 answer
532 views

An Exponential Sum Restricted to Primes

Let $a,q,N$ be integers such that $N/2 \leq q \leq N$ and $a/q \notin \mathbb{Z}$. Is the following estimate true, and, if so, how can it be proved? \[\left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \...
Linden's user avatar
  • 217
4 votes
1 answer
307 views

When the Kloosterman sum is an integer?

Let $q$ be a power of prime $p$ and $\zeta_p$ be the complex $p$ th root of unity. We denote by $\mathbb{F}_q$ the finite field of $q$ elements and by $Tr$ the absolute trace function $\mathbb{F}_q\...
sampath's user avatar
  • 255
4 votes
2 answers
366 views

On an observation which relates to the exponential sum $\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\frac{1}{2}+it}$

This observation is based on the numerical calculation of the exponential sum: $$\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\frac{1}{2}+it}$$ It is known that this sum is related to the famous Riemann–Siegel ...
Milin's user avatar
  • 395
4 votes
1 answer
299 views

The Wilton-type bounds involving half-integral weight cusp forms

There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following: Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be ...
hofnumber's user avatar
  • 563
4 votes
1 answer
332 views

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 ...
JACK's user avatar
  • 421
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 ...
Fatima Majeed's user avatar
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 ...
Joshua Stucky's user avatar
4 votes
0 answers
169 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 ...
Joshua Stucky's user avatar
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 ...
tomos's user avatar
  • 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}...
Mayank Pandey's user avatar
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 ...
Mayank Pandey's user avatar
3 votes
1 answer
338 views

Estimates for certain double-Kloosterman sums

Sorry to disturb. I encounter a double-Kloosterman sum, which needs some help from the experts here. For any $q\in \mathbb{N}^+$, how can we estimate the type of sum $$ \sideset{_{}^{}}{^{\ast}_{}}\...
hofnumber's user avatar
  • 563
3 votes
1 answer
229 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 ...
Kirill's user avatar
  • 416
3 votes
2 answers
367 views

How to estimate a mixed character sum $\sum_{h \in \mathbb{Z}/q \mathbb{Z}} \chi(f(h)) e(Ch/q)$?

Let $q = p^t$ where $p$ is prime. I am interested in estimating the complete exponential sum, which looks like $$ \sum_{0 \leq h < q} \chi( (h-a_1)(h-a_2)(h-a_3)) \ \bar{\chi}( (h-b_1)(h-b_2)(h-...
Johnny T.'s user avatar
  • 3,625
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 $\tau_3(n) = \sum_{d_1d_2d_3=n} 1$ ...
Joshua Stucky's user avatar
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)...
Johnny T.'s user avatar
  • 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.
Pablo's user avatar
  • 11.3k
3 votes
0 answers
341 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 $$...
user avatar
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,...
Turbo's user avatar
  • 13.9k
2 votes
1 answer
218 views

Moments of certain exponential sum

Let $v(\beta) := \sum_{n\le X} e(n\beta)$ where $e(\alpha) := e^{2\pi i \alpha}$. It is not hard to show that $$\log X\ll \int_0^1 |v(\beta)| d\beta\ll \log X$$ and by considering the underlying ...
Mayank Pandey's user avatar
2 votes
1 answer
237 views

Need some clarification to understand an inequality involving exponential sums

I was looking into Montgomery's proof of the large sieve inequality in his book on Topics in Multiplicative Number Theory, and on page $18$, we have $$B(x)=\sum_{k=-\infty}^{\infty}b_ke(kx),$$ for $x\...
Anish Ray's user avatar
  • 309
2 votes
1 answer
304 views

Exponential sum (linear in the argument) over primes

Suppose we have $\alpha \in \mathbb{R}$. Then we know that $$\sum_{1 \leq n \leq X} e(n \alpha) \ll \min \{ X, \|\alpha\|^{-1} \}$$ where $\| \cdot \|$ is the distance to the nearest integer. I ...
Johnny T.'s user avatar
  • 3,625
2 votes
1 answer
202 views

Exponential sums involving smooth truncated divisor functions

Let $p$ be a prime, $a \neq 0$ an integer, let $M,N \gg 1$ and let $\psi,\eta$ be some fixed Schwartz functions. Would you know of any references in the literature where upper bounds for sums such as $...
user152169's user avatar
2 votes
1 answer
161 views

Uniform power-saving estimate for an exponential sum

Let $N$ be a large natural number. Define an expoential sum $$ I_m=\sum_{x,y=1}^N e^{2\pi i\frac{x^2-y^2}{N}m}, \,\, m=1,2,...,N-1. $$ The trivial bound for $I_m$ is $N^2$, as there are $N^2$ terms. ...
Tony B's user avatar
  • 463
2 votes
1 answer
229 views

Upper bound for a higher dimensional Ramanujan sum

Fix an integer vector $\mathbf m\in \mathbb Z^k$. Let $q$ be a positive integer. Is there a "good" upper bound in terms of $q,\bf m$ for the exponential sum: \[\sum_{\mathbf n} e\left(\frac{\langle m,...
Changguang's user avatar