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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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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 ...
Madarb's user avatar
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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 $...
user152169's user avatar
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Incomplete Character/Exponential Sums

Incomplete Exponential Sums with multiplicative and non-multiplicative coefficients are very interesting objects in theory and applications. On a previous question Modified Gauss Sum when the ...
user524928's user avatar
3 votes
1 answer
592 views

Bounds for some exponential type sum

Let $m \in \mathbb{N}$. Given that $$\sum_{\frac{T}{\log^{2} T} < n < T} n^{iT} \ll T^{1/2}\log^{2} T,$$ is it true that $$\sum_{\frac{T}{\log^{2} T} < n < T, n \neq m} \frac{n^{iT}}{m^{1+\...
Reviewer 2.'s user avatar
2 votes
2 answers
219 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^{...
Christopher-Lloyd Simon's user avatar
3 votes
0 answers
199 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
0 answers
136 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
253 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
2 votes
0 answers
120 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 ...
tomos's user avatar
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1 vote
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\...
ABB's user avatar
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1 vote
1 answer
211 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)\...
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
224 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: $$...
hofnumber's user avatar
  • 553
0 votes
0 answers
90 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{_{}^{...
hofnumber's user avatar
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1 vote
0 answers
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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
341 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
494 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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2 votes
1 answer
149 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} \...
hofnumber's user avatar
  • 553
4 votes
1 answer
262 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
168 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 ...
user479223's user avatar
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6 votes
2 answers
641 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
126 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
11 votes
1 answer
781 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
1 answer
136 views

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
122 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(...
Itachi's user avatar
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1 vote
0 answers
62 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
10 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
0 votes
1 answer
183 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
132 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
467 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
167 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
  • 159
10 votes
1 answer
461 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}\...
Dapao Zhang's user avatar
6 votes
1 answer
179 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
92 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
  • 579
0 votes
1 answer
293 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
183 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
632 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
3 votes
1 answer
333 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
140 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
225 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
94 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
204 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
  • 1,226
1 vote
0 answers
121 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
245 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
138 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
126 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
316 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 $0$ ...
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