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.
16 questions from the last 365 days
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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
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1
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292
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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
$$
M(x)=\sum_{q\le x}\sum_{\substack{0\lt a\le ...
- 178
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What is the Hausdorff dimension of the set on which this exponential sum is bounded?
This is a direct follow up to For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
What is the Hausdorff dimension of the ...
- 6,155
13
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2
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800
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For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
Can we characterise the set of rationals $x$ for which the sum
$$A_N(x)\, :=\, \sum_{n = 0}^N e(2^n x)$$
remains bounded ...
- 6,155
6
votes
2
answers
425
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Average of gcd of sum of two $k$th powers
I am interested bounding the following quantity. Given fixed $k \in \mathbb{N}$, $a,b \in \mathbb{Z}$, $\sigma \in [0,1)$, and intervals $I_1,I_2 \subset \mathbb{Z}$ can we establish the bound
$$S = \...
1
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0
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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 $|...
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0
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70
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$\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
144
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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,...
- 301
2
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0
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191
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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 $$\sqrt x\ll \int _0^...
- 1,381
4
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1
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338
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Bounds on quadratic character sums
I asked this question on Mathematics stack exchange but didn't get a response, so I ask here too.
Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free ...
- 153
2
votes
1
answer
202
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Exponential sums involving smooth truncated divisor functions
Let $p$ be a prime, $a \neq 0$ an integer, let $M,N \gg 1$ and let $\psi,\eta$ be some fixed Schwartz functions. Would you know of any references in the literature where upper bounds for sums such as
$...
- 253
2
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2
answers
231
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Conditional convergence of exponential sums related to a Hecke modular form
Definition
Consider the Fourier coefficients $\psi(n)$ of the modular form $\eta^4(6\tau)$,
which are defined in terms of $q=\exp(i2\pi\tau)$ by the identity:
$$\eta^4(6\tau) = q \prod_1^\infty (1-q^{...
2
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0
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209
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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
- 67
4
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0
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154
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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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2
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2
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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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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|\...
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