Questions tagged [exponential-sums]
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 exponential sums is necessary for studying modern number theory.
215 questions
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Character and exponential sums over primes under GRH
Theorem 18.13 of Montgomery and Vaughan's multiplicative number theory book says for non-principal characters (ignoring $\log $'s) $$\int _{-\delta }^{\delta }|\psi _\chi (\beta )|^2d\beta \ll \delta ...
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1
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Variant of van der Corput's kth-derivative estimate
I ask this question as a non-expert in analytic number theory who needs to use a number theory result. Lemma 2.2 of this paper by Chan, Kumchev and Wierdl reads as follows:
Lemma 2: Let $k \ge 2$ be ...
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A system of T-functions on $(0,\infty)$
Let $r>1$ be real and
\begin{align*}
f_1(x) &= 1,\\
f_2(x) &= (x+4)^r,\\
f_3(x) &=(x+4)^r(x+3)^r,\\
f_4(x) &= (x+4)^r(x+3)^r(x+2)^r,\\
f_5(x) &=(x+4)^r(x+3)^r(x+2)^r(x+1)^r.
\...
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vote
1
answer
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Positive roots of real exponents function
Let $$f(x) = a+b(x+3)^r+c(x+3)^r(x+2)^r+d(x+3)^r(x+2)^r(x+1)^r,$$ where $r>1$ and $a,b,c,d$ are real numbers. I need to prove that $f(x)$ can't have more than $3$ positive zeros.
Attempts -
By ...
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0
answers
191
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What do know about the exceptional set in Goldbach under various hypothesis
it's known that under GRH the exceptional set in Goldbach is essentially $\ll \sqrt x$ (here and throughout I don't care for $\log $'s and $\epsilon $'s). Ultimately this comes from $$\psi _\chi (\...
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1
vote
1
answer
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Major arc approximations - partial summation and Gallagher's lemma
The motivation for my question is that I think I need to use Gallagher's lemma for exponential sums (``A large sieve density estimate near $\sigma =1$", Lemma 1, https://link.springer.com/article/...
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1
answer
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Dominating term in exponential polynomial
Let $$f(x) = \sum_{k=1}^{m}(-1)^k e^{-x\alpha_k} = -e^{-x\alpha_1}+e^{-x\alpha_2}-\cdots+(-1)^me^{-x\alpha_m}$$ be an exponential polynomial, where $\alpha_i$'s are distinct positive real numbers. Let ...
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votes
1
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Bounding a complicated sum with variable ceilings and floors in terms of $q$, $x$, $H$
I am studying sums that appear in the analysis of exponential sums and need a good big–O bound in terms of $(q, x, H)$. Specifically, let
$$
T = \frac{x}{\gamma q} + \frac{m}{\gamma}, \quad 1\le m \le ...
- 540
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0
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Mean square / Plancherel with Farey dissection
Suppose $S(\alpha )$ is the exponential sum up to $x$ of a sequence with mean square $x$. Trivially $\int _0^1|S(\alpha )|^2d\alpha \ll x$. Should it be true that $\sum _{q,a}\cdot \frac {q}{a}\cdot \...
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Proving that a certain factorial double sum collapses to a double-factorial series
While solving a few sums, I came across the following double sum
$$\sum_{k=0}^{n} (-1)^k\frac{(n+k)!}{2^kk!(n-k)!}x^{n-k}\sum_{j=0}^{n+k} \frac{x^j}{j!}$$
which is expected to evaluate to
$$\sum_{k=0}^...
3
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1
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Struggling with what should be an easy identity about quadratic exponential sums $\frac 1 q \sum_{k = 0}^{q-1} e_q(k^2a -km)$
I took the last few weeks to read material from other textbooks, but I decided to circle around to Appendix A and build off of my first question.
Let
$$S(a,m;q) = \frac 1 q \sum_{k = 0}^{q-1} e_q(k^2a ...
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1
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Question about the splitting step in Baker's proof in "Smooth Numbers in Beatty Sequences"
I'm reading Roger Baker's paper "Smooth Numbers in Beatty Sequences", and I'm trying to understand the argument starting on page 4, where he proves Theorem 2.
In that section, Baker ...
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votes
1
answer
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Secondary term in asymptotic formula for number of integral solution to quadratic form $F(x)=0$
In Heath-Brown's paper " A new form of the circle method and applications to Quadratic forms", Theorem 7 states that the number of weighted integral solutions in an expanding region $P\...
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votes
1
answer
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Lower bound for trilinear character sum in $\mathbb{Z}_2^n$
Consider the group $\mathbb{Z}_2^n$, equipped it with the following dot product: for $a=(a_1,\dots,a_n)\in \mathbb{Z}_2^n$ and $b=(b_1,\dots,b_n)\in \mathbb{Z}_2^n$, define $a\cdot b :=a_1b_1+\dots+...
- 448
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2
answers
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Quadratic Gauss sum with half the usual exponent
I am faced with the variant of the quadratic Gauss sum which has just half the usual exponent (a $\pi$ in place of $2\pi$). Explicit computer checks suggest that
$$
k \in 2 \mathbb{N}_{>0}
\;\;\...
- 21.1k
6
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1
answer
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Bounds for a 2D hyper Kloosterman sum
Let $a,b,c,q$ be positive integers. One way to generalize the standard Kloosterman sum to two variables is
$$
K(a,b,c;q) := \sum_{\substack{x_1\, \text{(mod $q$)}\\ (x_1,q)=1}} \sum_{\substack{x_2\, \...
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1
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Degree and Parity of Polynomials in the Expansion of $ g(T, z) $
I am reading (https://www.ams.org/journals/mcom/2011-80-274/S0025-5718-2010-02426-3/S0025-5718-2010-02426-3.pdf) and considering the function given on page 1004:
$
g(T,z) = \exp \left( -\left(d + \...
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Improving the correct Constant in Lemma 3 of "Explicit Estimates for the Riemann Zeta Function" by Cheng and Graham
I am studying quadratic exponential sums of the form $f(x) $ = $\alpha x^2$ + $ \beta x $ + $ \gamma $ and attempting to improve the constant in Lemma 3 of the paper:
Explicit estimates for the ...
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Literature about techniques of estimation of exponential sums
In mathematics there have been developed a lot of techniques connected with estimation of various exponential sums. However, I did not succeed in finding the literature(in English) which tells in ...
7
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Deriving inequality (8.9) from (8.8), in Iwaniec–Kowalski “Analytic Number Theory”
I am working through the problem presented in Chapter 8 of Iwaniec and Kowalski’s Analytic Number Theory (specifically inequalities (8.8) and (8.9)) and I am struggling with the transition between ...
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6
votes
1
answer
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Prime number theorem via large sieve type sums
We know that the prime number theorem is equivalent to the statement
$$
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 ...
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votes
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What is the Hausdorff dimension of the set on which this exponential sum is bounded?
This is a direct follow up to For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
What is the Hausdorff dimension of the ...
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For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
Can we characterise the set of rationals $x$ for which the sum
$$A_N(x)\, :=\, \sum_{n = 0}^N e(2^n x)$$
remains bounded ...
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6
votes
2
answers
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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
$$S = \...
1
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0
answers
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Period of the modulus of a complex exponential sum
Consider a sum of exponentials function of the integer $x$: $f(x)=A(x)+B(x)$, where $A(x)=\sum_{i=1}^n c_i \theta_i^x$ with $\theta_i$ roots of unity, and $B(x)=\sum_{j=1}^m d_j\lambda_j^x$ with $|...
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$\ell^2 \rightarrow L^p ([0,1]^d) $ estimates for trigonometric polynomials
My question concerns $L^p ([0,1]^d)$ estimates for trigonometric polynomials, where both the coefficients and frequencies are coming from general (i.e. not necessarily geometrically special/structured)...
2
votes
1
answer
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Exponential sums over a linear subspace
I'm looking into certain type of exponential sums, which are summed over a linear subspace, and I couldn't find a good reference for that.
The (simplified) setting is the following. Let $p$ be a prime,...
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0
answers
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The exponential sum over primes on average
In https://academic.oup.com/blms/article-abstract/20/2/121/266256?redirectedFrom=fulltext Vaughan shows the following bounds for the $L^1$-mean of the exponential sum over primes $$\sqrt x\ll \int _0^...
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Bounds on quadratic character sums
I asked this question on Mathematics stack exchange but didn't get a response, so I ask here too.
Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free ...
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votes
1
answer
251
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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
$...
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votes
2
answers
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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:
$$\eta^4(6\tau) = q \prod_1^\infty (1-q^{...
2
votes
0
answers
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A problem about the series $\sin(n^p)$ [closed]
Prove that when $p>0,$ the series $$\sum_{n=1}^\infty \sin(n^p)$$
is divergent
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votes
0
answers
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Vanishing exponential sums of fractional parts of polynomials
Let $p$ be an integer polynomial and $k$ be a natural number, both fixed. Is it the case that if
$$C(\alpha) = \sum_{i=1}^k e(\alpha \{p(i)/k\})$$
equals 0, then $\alpha$ is an integer? Here, $e(x) = ...
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If $\inf\{b\in\mathbb{R}\mid\sum_{n=1}^{\infty}e^{-ax_n-by_n}<+\infty\}=1-a$ for all $a\in [0,1]$, does this equality hold for all $a\in\mathbb{R}$?
Let $\left\{x_n\right\}_{n=1}^{+\infty},\left\{y_n\right\}_{n=1}^{+\infty}\subset [0,+\infty)$ be two sequences of non-negative real numbers. Suppose there exist $\lambda\ge 1, c\ge 0$ such that $\...
user519738
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what is the current best estimation for the upper bound of the exponential sum for an arbitrary irrational number $\alpha$
I would like to know what the current best estimation for the upper bound of the exponential sum
$$\left|\sum_{n=1}^N \exp \left(2 \pi i\alpha\left(x_0+x_1 n+\ldots+x_d n^d\right)\right)\right|=\left|\...
- 543
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0
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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 ...
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votes
0
answers
190
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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 ...
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The number of roots of pseudo-exponential polynomials
Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
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1
vote
1
answer
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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
$$
\sum_{q\le Q} \sum_{\substack{0\lt a \lt q \\ (a,q)=1}}|S_x(a/q)|^2 \le (Q^2+4\pi x)\...
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1
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Need some clarification to understand an inequality involving exponential sums
I was looking into Montgomery's proof of the large sieve inequality in his book on Topics in Multiplicative Number Theory, and on page $18$, we have
$$B(x)=\sum_{k=-\infty}^{\infty}b_ke(kx),$$ for $x\...
- 309
1
vote
1
answer
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On the estimate for a double exponential sum
I encounter a hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,s \in \mathbb{N}^+$(which may not be necessarily co-prime with each other), how to bound the sum:
$$...
- 573
0
votes
0
answers
118
views
A question on the evaluations of certain three-dimensional hyper-Kloostermans
There is a basic question regrading the 3-dimensional hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,h \in \mathbb{N}$, how to estimate the sum:
$$\sideset{_{}^{...
- 573
1
vote
0
answers
132
views
Manyfold iterated exponential sum with growing conductor
Let $\varphi(x_1,\dots, x_k)$ be some smooth function with partial derivatives of magnitude $\asymp 1$ for $x_i\asymp 1$. For concreteness, as it doesn't appear to add much extra structure beyond the ...
- 1,909
5
votes
1
answer
520
views
Exponential sum involving floor function
Can one get cancellation in exponential sums such as, say,
$$
\sum_{n\sim N} e(\lfloor n^\theta\rfloor^\beta),
$$
for fixed positive $\theta,\beta\not\in\mathbb Z$? When $\theta < 1$, it seems ...
- 1,909
7
votes
1
answer
666
views
Does there exist some irrational $x,\alpha$ so that this Weyl sum is $o(\sqrt N)$?
This is a less ambitious version of Is the Lebesgue measure of the $x$ so that this exponential sum is $o(\sqrt{N})$ positive? .
Consider $$S_N(x):=\sum_{n=1}^N \exp\left(2\pi i\left(\frac12n^2x+\...
- 2,345
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votes
1
answer
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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 ...
- 2,345
2
votes
1
answer
201
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
$$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p}
\...
- 573
4
votes
1
answer
372
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The Wilton-type bounds involving half-integral weight cusp forms
There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following:
Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be ...
- 573
0
votes
1
answer
249
views
Exponential sum with weight in bottom
I am interested in the exponential sum
$$\sum_{n=1}^X \frac{e(c_1n^2+c_2 n)}{1-e(c_1n)}$$
where $c_2$ is irrational and $e(x)=e^{2\pi i x}$. I know if the denominator is not there, this is a Weyl sum ...
- 2,345
6
votes
2
answers
727
views
Number of solutions of $am \equiv bn \pmod{q}$
Let $q$ be a (large) prime. Let $N$ be a positive integer of size${}\approx \sqrt{q}$. Let $\mathcal{M}$ be an arbitrary subset of $\{1, \dots, q\},$ such that $\mathcal{M}$ has cardinality $N$. ...
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