All Questions
80 questions
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
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
$$
M(x)=\sum_{q\le x}\sum_{\substack{0\lt a\le ...
- 178
6
votes
2
answers
425
views
Average of gcd of sum of two $k$th powers
I am interested bounding the following quantity. Given fixed $k \in \mathbb{N}$, $a,b \in \mathbb{Z}$, $\sigma \in [0,1)$, and intervals $I_1,I_2 \subset \mathbb{Z}$ can we establish the bound
$$S = \...
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 $$\sqrt x\ll \int _0^...
- 1,381
2
votes
1
answer
202
views
Exponential sums involving smooth truncated divisor functions
Let $p$ be a prime, $a \neq 0$ an integer, let $M,N \gg 1$ and let $\psi,\eta$ be some fixed Schwartz functions. Would you know of any references in the literature where upper bounds for sums such as
$...
- 253
4
votes
0
answers
78
views
Repeated values of a monomial
Let $H,M\geq 1$ and let $h_0$ and $m_0$ be fixed integers with $(h_0,m_0)\in [H,2H]\times[M,2M]$. Let $\alpha$ be a positive real number. I'm trying to find an upper found for the number of integer ...
- 1,990
2
votes
0
answers
143
views
The exponential sum of $\omega (n)$
Let $\omega (n)$ be the number of (distinct) prime divisors of $n$ $$\omega (n)=\sum _{p|n}1$$ and let $S(a/q)$ be its exponential sum $$\sum _{n\leq x}\omega (n)e(na/q).$$
Question 1: Can anyone give ...
- 1,381
1
vote
1
answer
244
views
Large sieve type inequality
Let $S_x(t)=\sum_{n\le x} a_n e(nt)$, where $e(x)=e^{2\pi i x}$. Then, the large sieve inequality tells us that
$$
\sum_{q\le Q} \sum_{\substack{0\lt a \lt q \\ (a,q)=1}}|S_x(a/q)|^2 \le (Q^2+4\pi x)\...
- 178
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\...
- 309
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
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 ...
- 1,890
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 $$S_N(x):=\sum_{n=1}^N \exp\left(2\pi i\left(\frac12n^2x+\...
- 1,904
6
votes
1
answer
283
views
Is the Lebesgue measure of the $x$ so that this exponential sum is $o(\sqrt{N})$ positive?
Consider $$S_N:=\sum_{n=1}^N \exp\left(2\pi i\left(\frac12n^2x+\alpha n\right)\right)$$
where $\alpha$ is irrational. For certain $x$ (say integer) we can get that this is bounded for all $N$. I am ...
- 1,904
4
votes
1
answer
299
views
The Wilton-type bounds involving half-integral weight cusp forms
There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following:
Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be ...
- 563
2
votes
0
answers
128
views
Distribution of square roots (mod m) on small intervals (with respect to m)
Fix a large positive integer $m$. Let $A$ be small positive number typically $\sim m^{1/3}$. Suppose $S(A, m)$ be set of solutions (normalized by dividing by $m$) to the quadratic congruences $x^2 = a ...
- 577
1
vote
1
answer
130
views
Bound for some trigonometric polynomials
Let $e(x)=e^{2\pi i x}$ and consider the following functions defined for $x\in [0,1]$:
$$
f_1(x)=\frac{e(10x)-e(x)}{e(x)-1}, \quad f_2(x)=\frac{e(110x)-e(11x)}{e(11x)-1},
$$
and
$$
f_3(x)=\frac{e(...
- 178
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
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}\...
- 351
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 ...
- 1,990
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
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
$$\sum_{m\le M} \min \left(N,...
- 563
3
votes
1
answer
338
views
Estimates for certain double-Kloosterman sums
Sorry to disturb. I encounter a double-Kloosterman sum, which needs some help from the experts here.
For any $q\in \mathbb{N}^+$, how can we estimate the type of sum
$$ \sideset{_{}^{}}{^{\ast}_{}}\...
- 563
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 ...
- 1,990
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
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
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
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
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
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
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 ...
- 353
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 ...
- 421
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
$$
\sum_{(a_1, a_2, a_3, q) = 1} \sum_{\mathbf{h} \in (\mathbb{Z}/q\mathbb{Z})^n }e( \frac{a_1}{q}\ell_1(h_1, \ldots, h_n)...
- 3,625
5
votes
0
answers
302
views
Exponential sums with prime power modulus
I am looking for an analogue of the following result of Fouvry and Katz for prime power modulus ("A general stratification theorem for exponential sums, and applications", J. reine angew. Math. 540 (...
- 26.9k
5
votes
0
answers
124
views
Linear exponential sum with gcd
The sum $$\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.$$ Indeed with a standard estimate for a linear exponential sum it ...
- 143
4
votes
0
answers
169
views
Smoothed Weyl sum inequality
One version of Weyl's inequality states that for any $\alpha\in\mathbb{R}$ and $(a, q) = 1$ such that $|\alpha - a/q|\le 1/q^2$, we have that
$$\sum_{n\le X} e(n^k\alpha)\ll X^{1 + \varepsilon}(q^{-1}...
- 1,890
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 $N$
$$
\left| \sum_{x \in ...
- 334
1
vote
0
answers
56
views
Discrepancy bound of integer tensor product sequence?
Here discrepancy is from $(2.4)$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The (extreme) discrepancy $D_N(P) = D_N(x_l,\dots,X_N)$ of
the point set $P$ of $N$ points in $...
- 13.9k
3
votes
0
answers
206
views
Cancellation in this exponential sum?
I would like to know whether it is possible to obtain cancellation in the sum
$$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$
where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.
- 11.3k
2
votes
1
answer
161
views
Uniform power-saving estimate for an exponential sum
Let $N$ be a large natural number.
Define an expoential sum
$$
I_m=\sum_{x,y=1}^N e^{2\pi i\frac{x^2-y^2}{N}m}, \,\, m=1,2,...,N-1.
$$
The trivial bound for $I_m$ is $N^2$, as there are $N^2$ terms. ...
- 463
2
votes
1
answer
218
views
Moments of certain exponential sum
Let $v(\beta) := \sum_{n\le X} e(n\beta)$ where $e(\alpha) := e^{2\pi i \alpha}$. It is not hard to show that
$$\log X\ll \int_0^1 |v(\beta)| d\beta\ll \log X$$
and by considering the underlying ...
- 1,890
6
votes
1
answer
287
views
Number of solutions for the inequality with square roots
Let $M$ be some large real number and $\delta>0$. I would like to estimate the number of solutions for the inequality
$$|\sqrt{n_1}+\sqrt{n_2}-\sqrt{n_3}-\sqrt{n_4}|<\delta\sqrt{M},$$
where $...
- 7,193
3
votes
2
answers
367
views
How to estimate a mixed character sum $\sum_{h \in \mathbb{Z}/q \mathbb{Z}} \chi(f(h)) e(Ch/q)$?
Let $q = p^t$ where $p$ is prime. I am interested in estimating the complete exponential sum, which looks like
$$
\sum_{0 \leq h < q} \chi( (h-a_1)(h-a_2)(h-a_3))
\ \bar{\chi}( (h-b_1)(h-b_2)(h-...
- 3,625
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
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:
$$\sum_{N\leq n\leq 2N}\exp(...
4
votes
2
answers
366
views
On an observation which relates to the exponential sum $\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\frac{1}{2}+it}$
This observation is based on the numerical calculation of the exponential sum:
$$\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\frac{1}{2}+it}$$
It is known that this sum is related to the famous Riemann–Siegel ...
- 395
1
vote
0
answers
163
views
Is this averaged exponential sum over primes small infinitely often?
Do there exist infinitely many positive integers $N$ such that
$$\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