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.
196 questions
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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
$$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p}
\...
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On the upper-bound for a type of quintuple Kloosterman sums
Sorry to disturb, dear experts here. I have a question involving the quintuple Kloosterman sum, and expect some hints to show the upper-bound.
My question is, for any $x,y,z,w,\delta \in \mathbb{Z}$ ...
- 563
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Moments of certain exponential sum
Let $v(\beta) := \sum_{n\le X} e(n\beta)$ where $e(\alpha) := e^{2\pi i \alpha}$. It is not hard to show that
$$\log X\ll \int_0^1 |v(\beta)| d\beta\ll \log X$$
and by considering the underlying ...
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Conditional convergence of exponential sums related to a Hecke modular form
Definition
Consider the Fourier coefficients $\psi(n)$ of the modular form $\eta^4(6\tau)$,
which are defined in terms of $q=\exp(i2\pi\tau)$ by the identity:
$$\eta^4(6\tau) = q \prod_1^\infty (1-q^{...
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Integral over an exponential sum with squares
How should I estimate the following integral
$$I = \int_0^1 \left( \sum_{n=0}^{p-1} e(n^2t) \right)^2 dt $$
where $p$ is a prime?
Here is the method I followed:
\begin{align*}
I & = \int_0^1 \...
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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\...
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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 $\...
- 605
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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 ...
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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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A conjecture relating an integral and a sum, the floor function and squares
I've found through evidence and have conjectured on a math publication that:
$$\Big\lfloor\int_1^\infty (k^{1/(k^{1+1/\sqrt{x}})} - 1)dk\Big\rfloor = \Big\lfloor\sum_{k=1}^{\infty}k^{1/(k^{1+1/\sqrt{x}...
user475930
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Uniform power-saving estimate for an exponential sum
Let $N$ be a large natural number.
Define an expoential sum
$$
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. ...
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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,...
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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 $m,n\in\mathbb{Z},\;q,c\in\mathbb{N}$.
...
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Elaboration of a certain section of a paper by Thanigasalam
In section 11 of this paper by Thanigasalam, it says "... we get $G(10)\le 105$, and this implies that $H(10) \le 107$". However, it is very unclear how this follows. Why is it the case that $G(10)\le ...
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Proof of Szegö asymptotic theorem
Consider the truncated exponential series
$$P_N(z) = \sum_{n= 0}^N \frac{z^n}{n!}$$
The zeros of this series have been studied by Szëgo and others (see e.g. here). He established an asymptotic for the ...
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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 ...
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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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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
$$...
- 167
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Occurrence of simultaneous small remainders?
Fix $(a,b)=1$, $a<b<2a$ and $a,b>n^{1/(2t)}$ and fix a prime $T\approx n^{\tau+\frac1k}$ where $\tau\geq1$ and $k=2(t-1)$. We can show using exponential sums there is an $m_{_T}$ such that $T/...
user94040
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The exponential sum over primes on average
In https://academic.oup.com/blms/article-abstract/20/2/121/266256?redirectedFrom=fulltext Vaughan shows the following bounds for the $L^1$-mean of the exponential sum over primes $$\sqrt x\ll \int _0^...
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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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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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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 ...
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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
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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.
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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^...
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Converse of Gallagher identity
A well known useful inequality of Gallagher states (in one form) that for any sequence $a:\mathbb N\to\mathbb C$, we have that $$\int_{|\theta|\le\delta} \bigg|\sum_n a(n)e(n\theta)\bigg|^2d\theta\ll
\...
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An exponential sum like the Kloosterman sums
I encounter a tricky sum like the Kloosterman sum
$${\sum_{x \mod P}}^\ast e\left(\frac{ax+\overline{x}}{P}+\frac{lx^2}{P^2}\right),$$
where $l$ is a positive integer co-prime with $P$ and here $P$ ...
- 353
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An exponential sum estimate on small intervals
Let $1<r<2$ be a real number. Let $4<p\le 6$. Consider the exponential sum estimate
$$\int_0^{2\pi}\int_0^{N^{r-2}} \left|\sum_{n=1}^N e^{inx+in^2 y}\right|^p \, dy \, dx$$
Notice that the $y$...
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Convergence of a tetration series
Let $0<a\neq 1$ be a fixed real number and denote by $a^{ \frac{x}{}}:=\mbox{uxp}_a(x)$ the ultra exponential function (ultra power) that is a unique extension of tetration (and also its linear ...
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Can the matrix exponential be equal to the elementwise exponential [closed]
Just out of curiosity: does there exist a matrix $A=(a_{i,j}) \in \mathbb{C}^{n\times n}, n>1$ such that $(e^{a_{i,j}})\in \mathbb{C}^{n\times n}$ is equal to the matrix exponential $e^A=\sum_{k\...
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On exponential sum weighted with von-Mangoldt function
Suppose we have $\alpha \in \mathbb{R}$ such that $|\alpha - a/q| < 1/q^2$,
where $(a,q)=1$. Then we know that the exponential sum
$$
S(\alpha) = \sum_{1 \leq n \leq X} \Lambda(n) e(n \alpha)
$$
...
- 3,625
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Kloosterman-like sum with inverse to different moduli
In some recent work, the following strange-looking exponential sum arose:
$$
\sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg).
$$
Here $e(x) = e^{2\pi i x}$ as ...
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Large sieve type inequality
Let $S_x(t)=\sum_{n\le x} a_n e(nt)$, where $e(x)=e^{2\pi i x}$. Then, the large sieve inequality tells us that
$$
\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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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 $q\in \mathbb{N}^+$,...
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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,...
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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 ...
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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:
$$...
- 563
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Coefficients in the sum $\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{j,m}(n-k)^jk^j=n^{2m+1}, \ m=1,2,....$
Consider a sum $$\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{j,m}(n-k)^jk^j$$
which returns an odd power $n^{2m+1}$ of $n$, for $\ m=0,1,2,...$ given fixed $A_{0,m}, \ A_{1,m}, \ ..., \ A_{m,m}$. The ...
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Coefficients $U_m(n,k)$ in the identity $n^{2m+1}=\sum\limits_{0\leq k \leq m}(-1)^{m-k}U_m(n,k)\cdot n^k$
Review the main result of mathoverflow.net/questions/297900, that is the identity
\begin{equation}\label{f1}
n^{2m+1}=\sum\limits_{1\leq k \leq n}\sum\limits_{j\geq0}A_{m,j}k^j(n-k)^j,
\end{equation}
...
- 167
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Uniform convergence of infinite sum with Dirichlet characters
I would like to prove uniform convergence of function series like :
$$\sum\limits_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in ...
- 1,199
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stationary phase method in analytic number theory
I hope someone can tell me something about the error term in the formula calculating the oscillatory integral like $\int_a^b g(x)e(f(x))d x$. Specially, the exact formula on page 114 of M. Huxley's ...
- 177
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1
answer
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The number of roots of pseudo-exponential polynomials
Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
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Bound for some trigonometric polynomials
Let $e(x)=e^{2\pi i x}$ and consider the following functions defined for $x\in [0,1]$:
$$
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
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1
answer
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Exponential sum estimates similar to the one for $\sum_p (\log p) e(p \alpha)$, but for different sequences
Obtaining a non-trivial estimate for $\sum_p (\log p) e(p \alpha)$ over the minor arcs is one of the estimates required for obtaining the ternary Goldbach for $n$ sufficiently large via the circle ...
- 579
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2
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A double sum with complex numbers having stochastic variables
I am very confused by a sum I have been trying to solve analytically/ numerically for a long time. It comes from the idea of a physical problem where the observation is made that has a combined ...
- 159
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1
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Distribution of quadratic polynomials mod $n$ and $n^2$
Suppose $n$ is odd, then both equations $x^2 = D \; mod \;n$ and $x^2 = D \; mod \;n^2$ have the same number of solutions for fixed $D$ coprime to $n$. What can be said about the relationship between ...
- 577
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An inequality between sum of exponential functions wrt dyadic index
I am reading a paper 'Periodic Nonlinear Schrodinger Equation and Invariant Measures' written by J.Bourgain. And I am wondering if I can have some help from this website.
My question is an inequality ...
- 239
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answer
225
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Upper bound for the integral over minor arcs of the exponential sum with prime omega function coefficients
Define $\mathfrak{m}$ as the union of the minor arcs of the form $|\alpha-\frac{a}{q}|\leq 1/qQ$, with $(a,q)=1$ and $Q_0<q\leq Q$, with $Q_0\geq N/Q$, for a certain $N\geq Q$ large.
Is it ...
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1
answer
239
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Simplify a equation
I have a problem simplifying the summation here:
$$
\sum_{x=0}^{n}\sum_{y=0}^{x} {n\choose{x}} {x\choose{y}} y!(x-y)!
$$
The last three terms can be simplified to x!, so the current summation ...
- 11