All Questions
Tagged with exponential-sums nt.number-theory
134 questions
4
votes
1
answer
465
views
Explicit numbers with square root cancellation in Weyl's exponential sum
I'm interested in examples of real numbers $\alpha$ where we have
$$\left| \sum_{n=1}^N \mathrm e(\alpha n) \right| \ll N^{1/2} $$
or perhaps with the weaker estimate with the right side replaced ...
- 41
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
154
views
Vanishing exponential sums of fractional parts of polynomials
Let $p$ be an integer polynomial and $k$ be a natural number, both fixed. Is it the case that if
$$C(\alpha) = \sum_{i=1}^k e(\alpha \{p(i)/k\})$$
equals 0, then $\alpha$ is an integer? Here, $e(x) = ...
- 41
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
$$\sum_{n\le X} e(n^k\alpha)\ll X^{1 + \varepsilon}(q^{-1}...
- 1,890
4
votes
0
answers
158
views
Unique factorization for the semigroup generated by {2cos(π/n) | n>3}?
Let $S$ be the multiplicative semigroup of numbers generated by $B=\{ 2cos(\frac{\pi}{n}) \mid n \ge 4 \}$.
Question: Does every number of $S$ factorize uniquely (up to perm.) as a product of ...
4
votes
0
answers
562
views
Best known bounds on certain exponential sums
What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...
- 1,890
3
votes
1
answer
338
views
Estimates for certain double-Kloosterman sums
Sorry to disturb. I encounter a double-Kloosterman sum, which needs some help from the experts here.
For any $q\in \mathbb{N}^+$, how can we estimate the type of sum
$$ \sideset{_{}^{}}{^{\ast}_{}}\...
- 563
3
votes
2
answers
326
views
Proof of the identity $\left\lvert\sum_{j=0}^{p-1}e^{(2\pi i/p)(mj^2+nj)}\right\rvert=\sqrt p$ for $p$ odd prime
While studying themes related to mutually unbiased bases, I've come across the following identity:
$$\left\lvert\sum_{j=0}^{p-1}e^{(2\pi i/p)(mj^2+nj)}\right\rvert=\sqrt p,$$
for $p$ odd prime and $m\...
- 342
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 ...
- 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
$$
\sum_{0 \leq h < q} \chi( (h-a_1)(h-a_2)(h-a_3))
\ \bar{\chi}( (h-b_1)(h-b_2)(h-...
- 3,625
3
votes
1
answer
382
views
Twisting by a multiplicative Character in Katz, Perversity and Exponential sums
Let $C(x_1,\ldots,x_n)$ be a nonsigular cubic form with integral coefficients.
In his proof that $C$ fulfills the Hasse-Principle, if $n\geq 9$, Hooley used the following estimate that was provided by ...
- 33
3
votes
1
answer
171
views
Exponential sums over rings
I'm trying to evaluate an exponential sum of the form:
\begin{equation}
\sum_{c\in Z_q}\chi(f(c))
\end{equation}
For polynomial $f(x)=a_2x^2+a_1x+a_0$ (with $a_2\ne 0$).
If $q$ is prime, then this is ...
- 301
3
votes
0
answers
82
views
growth rate of quadratic exponential sums with monomial coefficients
What is the growth rate of $$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{\...
- 301
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}}$$
where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.
- 11.3k
3
votes
0
answers
289
views
Prerequisites to read Katz's Gauss Sums, Kloosterman Sums, and Monodromy Groups
What is the minimal background one must have to read Katz's Gauss Sums, Kloosterman Sums, and Monodromy Groups?
- 1,890
3
votes
0
answers
340
views
On discrepancy of integer sequences related to Erdos-Turan-Koksma
Assume we have a sequence $a_1,\dots,a_k\in\Bbb N$ with each $a_i\approx n$ where $n$ is some integer.
Suppose there exists an $\eta\in(0,1)$ such that for every $d_1,\dots,d_k\in\Bbb Z$ with $$...
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
1
answer
154
views
On the estimate for the mixed 3-dimensional hyper-Kloosterman sum
There is a basic question regrading the mixed 3-dimensional hyper-Kloosterman sum:
For any positive integer $n$ not divisible by $p$, how to prove
$$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p}
\...
- 563
2
votes
1
answer
189
views
On the upper-bound for a type of quintuple Kloosterman sums
Sorry to disturb, dear experts here. I have a question involving the quintuple Kloosterman sum, and expect some hints to show the upper-bound.
My question is, for any $x,y,z,w,\delta \in \mathbb{Z}$ ...
- 563
2
votes
1
answer
218
views
Moments of certain exponential sum
Let $v(\beta) := \sum_{n\le X} e(n\beta)$ where $e(\alpha) := e^{2\pi i \alpha}$. It is not hard to show that
$$\log X\ll \int_0^1 |v(\beta)| d\beta\ll \log X$$
and by considering the underlying ...
- 1,890
2
votes
2
answers
231
views
Conditional convergence of exponential sums related to a Hecke modular form
Definition
Consider the Fourier coefficients $\psi(n)$ of the modular form $\eta^4(6\tau)$,
which are defined in terms of $q=\exp(i2\pi\tau)$ by the identity:
$$\eta^4(6\tau) = q \prod_1^\infty (1-q^{...
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
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 ...
- 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
$...
- 253
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
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,...
- 106
2
votes
1
answer
304
views
Kloosterman sum
Does anybody know the non-trivial bound for this sum?
$S(m,n,c,q)=\sum_{a,b\in \mathbb{Z}/cq\mathbb{Z}, \;ad\equiv 1\text{ mod }c} e^{2\pi i(am+nd)/qc},$
where $m,n\in\mathbb{Z},\;q,c\in\mathbb{N}$.
...
- 21
2
votes
2
answers
763
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 ...
- 579
2
votes
1
answer
144
views
Exponential sums over a linear subspace
I'm looking into certain type of exponential sums, which are summed over a linear subspace, and I couldn't find a good reference for that.
The (simplified) setting is the following. Let $p$ be a prime,...
- 301
2
votes
1
answer
228
views
Upper bound for an exponential sum in Waring-Goldbach problem
In Waring's problem, we have Hua's estimate
$$S(a,b,q) = \sum_{x=1}^q e^{2\pi i (ax^k + bx)/q)} \ll q^{1/2+\epsilon} \gcd(b,q),$$
where $(a,q)=1$.
?Do you know a similar upper bound for the sum
$$...
- 167
2
votes
2
answers
725
views
Occurrence of simultaneous small remainders?
Fix $(a,b)=1$, $a<b<2a$ and $a,b>n^{1/(2t)}$ and fix a prime $T\approx n^{\tau+\frac1k}$ where $\tau\geq1$ and $k=2(t-1)$. We can show using exponential sums there is an $m_{_T}$ such that $T/...
user94040
2
votes
0
answers
191
views
The exponential sum over primes on average
In https://academic.oup.com/blms/article-abstract/20/2/121/266256?redirectedFrom=fulltext Vaughan shows the following bounds for the $L^1$-mean of the exponential sum over primes $$\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
219
views
Is this limit zero?
Define $e(\theta)=e^{2\pi i\theta}, \theta\in [0,1]$, $P_n=\{p_1,p_2,...,p_n\}$ are the first $n$ primes, $\|f\|_1=\int_{[0,1]}|f(\theta)|d\theta$.
Problem 1.
is it true for all fixed $m\in \mathbb{N^...
- 543
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
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
$$
\sum_{q\le Q} \sum_{\substack{0\lt a \lt q \\ (a,q)=1}}|S_x(a/q)|^2 \le (Q^2+4\pi x)\...
- 178
1
vote
1
answer
237
views
A question involving the three-dimensional Kloosterman sum
Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.
For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$,...
- 563
1
vote
1
answer
346
views
On a sum like Kloosterman sum
I encounter a tricky sum like the Kloosterman sum
$$\sum_{x \mod qP} e ( \frac{x+\overline{x+P}}{qP} ),$$
where $q$ is a positive integer, $P$ is a prime number satisfying $(q,P)=1$, $x \bmod qP,(x,...
- 353
1
vote
3
answers
284
views
Decidability of sum of powers exponential diophantine equation
I want to ask about decidability of exponential Diophantine equation:
$z_12^{\eta_1} + \ldots + z_n2^{\eta_n} = z$, where $z_i,\eta_i,z \in \mathbb{Z}$, and $\eta_i$ - are variables.
Can we find ...
- 11
1
vote
1
answer
259
views
On the estimate for a double exponential sum
I encounter a hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,s \in \mathbb{N}^+$(which may not be necessarily co-prime with each other), how to bound the sum:
$$...
- 563