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.

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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^{...
Christopher-Lloyd Simon's user avatar
3 votes
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190 views

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
adobereader's user avatar
4 votes
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120 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) = ...
Borys Kuca's user avatar
2 votes
2 answers
246 views

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 $\...
YC Su's user avatar
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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|\...
katago's user avatar
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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 ...
Joshua Stucky's user avatar
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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 ...
tomos's user avatar
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1 vote
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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\...
ABB's user avatar
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1 vote
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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)\...
Itachi's user avatar
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2 votes
1 answer
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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\...
Anish Ray's user avatar
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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: $$...
hofnumber's user avatar
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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{_{}^{...
hofnumber's user avatar
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1 vote
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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 ...
Mayank Pandey's user avatar
4 votes
1 answer
319 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
7 votes
1 answer
480 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
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6 votes
1 answer
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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
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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} \...
hofnumber's user avatar
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4 votes
1 answer
238 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
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0 votes
1 answer
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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 ...
user479223's user avatar
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6 votes
2 answers
636 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 $N$. ...
Kurisuto Asutora's user avatar
2 votes
0 answers
124 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 ...
Melanka's user avatar
  • 577
10 votes
1 answer
716 views

Cancellation in a very rapidly oscillating exponential sum

Let $f(x)$ be a function such that for all $T \leq f(x), \ T / x \to \infty$ we have $$ \sum_{x \leq n < 2 x} e^{2 \pi i T / n} = o(x). $$ How fast can $f(x)$ grow? I can show that for any $\...
Random's user avatar
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2 votes
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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 ...
TheStudent's user avatar
1 vote
1 answer
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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(...
Itachi's user avatar
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1 vote
0 answers
61 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 ...
Joshua Stucky's user avatar
9 votes
1 answer
1k views

Why are Deligne-type exponential sum estimates so hard to use?

Let $F$ by a finite field, and $R(x_1,x_2,\ldots,x_n) := r_1(x_1,x_2,\ldots,x_n)/r_2(x_1,x_2,\ldots,x_n)$ a rational function in $n$ variables, frequently in analytic number theory or harmonic ...
Mark Lewko's user avatar
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0 votes
1 answer
182 views

Uncorrelation of exponential sums generated by irrational rotations over disjoint sets of integers

Assume that $\mathbb{N}=\{0,1,2,\ldots\}$ is partitioned into $k\ge 2$ disjoint sets $J(1),\ldots,J(k)$ such that for every $1\le p \le k$ the set $J(p)$ has an asymptotic density $$ d(J(p))=\lim_{n\...
Dominik Kwietniak's user avatar
1 vote
0 answers
126 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. ...
Hugo Chapdelaine's user avatar
4 votes
1 answer
407 views

Exponential sum vs. exponential integral via Poisson summation

When we want to estimate an exponential sum $$ \sum_{M<m\le M'}e(f(m)) \quad\text{with}\quad 1\le M\le M'\le 2M \quad\text{and}\quad e(x):=\exp(2\pi ix) $$ where $e(x):=\exp(2\pi ix)$ and the phase ...
snufkin26's user avatar
  • 343
1 vote
2 answers
162 views

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 ...
CfourPiO's user avatar
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10 votes
1 answer
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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
6 votes
1 answer
174 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
1 vote
0 answers
90 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 ...
SJY's user avatar
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0 votes
1 answer
265 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,...
hofnumber's user avatar
  • 553
2 votes
1 answer
176 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 $x,y,z,w,\delta \in \mathbb{Z}$ ...
hofnumber's user avatar
  • 553
14 votes
1 answer
613 views

The first case of the strong Littlewood conjecture

Let $A$ be a set of $n$ integers and consider the quantity: $$\int_{0}^1 \left| \sum_{a \in A} e^{2\pi i a x} \right|dx. $$ The (now solved) Littlewood conjecture is the claim that this quantity is ...
Mark Lewko's user avatar
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3 votes
1 answer
331 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
  • 553
4 votes
0 answers
134 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
1 vote
1 answer
220 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 $q\in \mathbb{N}^+$,...
hofnumber's user avatar
  • 553
5 votes
0 answers
91 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
1 vote
0 answers
193 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 ...
tomos's user avatar
  • 1,096
1 vote
0 answers
119 views

Self-referencing recurrence relation with exponential

I have the self-referencing recurrence relation $$ d(0) = 0 $$ $$ d(1) = a $$ $$ d(n+1) = d(n) + a*e^{-(\frac{d(n)-n*c}{b})^4} $$ Written as a sum: $$ d(N) = a*\sum_{n=0}^{N}e^{-(\frac{d(n)-n*c}{b})^...
Matte's user avatar
  • 11
4 votes
1 answer
231 views

Bound for sum of multiplicative character calculated over multivariate polynomial

Let $f \in \mathbb{F}_q[x_1, \dots, x_k]$ be a polynomial with $\deg f = n$, and let $\chi$ be a multiplicative character over $\mathbb{F}_q$. Is there any known bound, possibly with conditions about $...
José's user avatar
  • 209
1 vote
0 answers
136 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}...
Tony419's user avatar
  • 401
1 vote
1 answer
125 views

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 ...
Melanka's user avatar
  • 577
0 votes
1 answer
314 views

The exploration of the asymptotic behavior of a simple sum. $\sum_{k=1}^{\infty} (k^{1/k} - 1)$ [closed]

An informal investigation of a sum. Consider this sum: $$S =\sum_{k=2}^{\infty}(k^{1/k} -1)$$ Does this converge? How does it behave as it diverges, if it diverges? If $k$ equaled $1$ we would get $0$ ...
user avatar
2 votes
1 answer
374 views

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}...
user avatar
8 votes
1 answer
549 views

Why are exponential sums so bad at solving this very easy problem?

Exponential sums are a powerful tool in additive combinatorics and number theory. In my understanding, when it comes to estimate the cardinality of a certain set, exponential sums are (essentially) ...
locra's user avatar
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4 votes
0 answers
212 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
  • 1,096
6 votes
0 answers
104 views

Bounds on exponential and character sums of ratio of linear recurrences

Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\chi$ be a non-trivial additive character of $\mathbb{F}_q$, and let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$. Also,...
Erik4's user avatar
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