23
votes
Accepted
A finite alternating sum
I have obtained a formula for the generating function of your sequence.
Let $S_n$ be defined as in the quesion. We extend the definition to $n = 0$ by demanding $0^0 = 0$, hence $S_0 = 0$.
Consider $...
18
votes
A finite alternating sum
WhatsUp's generating function allows to get an asymptotics: we have $$\sum S_n t^n=\frac{t}{e^{t/e}-t},\sum S_n e^nt^n=\frac{t}{e^{t-1}-t}.$$
Denote $t-1=x$, then $$\frac{t}{e^{t-1}-t}=\frac{1+x}{e^x-...
17
votes
exponential sum over variety
This is a very general sort of problem and you will get very different answers depending on exactly which $V, f,g$ you need.
The least savings you could ask for is $\sqrt{p}$ savings over the ...
17
votes
Accepted
Deligne's theorem on exponential sums
Yes, smoothness is equivalent to the gradient being nonzero for every $x \in \overline{\mathbb F}_q^n \setminus \{0\}$.
I would define smoothness of the hypersurface defined by $Q_d$ as the condition ...
16
votes
Accepted
Why are Deligne-type exponential sum estimates so hard to use?
There are a lot of subtle reasons such exponential sums can fail to exhibit square-root cancellation. First let me comment on two reasons suggested in your answer:
(1) trying to have an explicit ...
13
votes
Accepted
Arguments of exponential sums
There are at least 3 ways of seeing this - an elementary way, via the determinants of $\ell$-adic sheaves, and via Katz's calculus of finite field hypergeometric functions.
Elementary:
Subtraction ...
11
votes
Mean square estimate for the Kloosterman sums
You cannot have estimates like (*) for any $\theta<1$. Fouvry and Michel showed that (see Theorem 1.2 there)
$$
\sum_{c\le x} |S(m,n;c)|/\sqrt{c} \gg_k \frac{x}{\log x} (\log \log x)^k,
$$
for ...
10
votes
Accepted
Does anyone recognize this exponential sum?
Yes, these sums occur as the arithmetic part of the Fourier expansion of
period kernels $\sum_{ad-bc=1}(a\tau+b)^{-k}(c\tau+d)^{-k}$, the analytic part
being J-Bessel functions. The derivation is not ...
10
votes
Accepted
The exploration of the asymptotic behavior of a simple sum. $\sum_{k=1}^{\infty} (k^{1/k} - 1)$
The OP asks the question whether the series $S =\sum_{k=2}^{\infty}(k^{1/k} -1)$ diverges.
The answer is that the series diverges, because $$k^{\frac{1}{k}}-1 = \exp\left(\frac{\ln k}{k}\right)-1>\...
10
votes
A basic estimate of exponential sums
The main idea is to apply Poisson summation formula. It allows to replace $a/q$ by $\lfloor q/a\rfloor$ and then repeat the procedure using continued fraction expansion of $a/q$. This idea belongs to ...
9
votes
Accepted
An exponential sum over squares
Actually, $|\sum_{M < n \leq N} e(x/n^2)| \sim c \sqrt{x}$, where $c \approx 0.016151690 + 0.0738060263i$.
To see this, write $$\sum_{M < n \leq N} e(x/n^2) = \sum_{M < n \leq \epsilon \sqrt{...
9
votes
counting points on unit sphere mod p
Since this old question has resurfaced, let me sketch two ways to prove the stated formulæ using algebraic geometry:
The first way is fairly elementary. Let us stick for definiteness with the number ...
9
votes
Upper bound an integral with exponential function
The integral in question can be rewritten as
$$
\begin{aligned}
I&:=\frac1{\sqrt n}\,\int_{-a\sqrt n}^{(1-a)\sqrt n} e^{-u^2}\Big(1-\exp\Big\{-\frac{u^4/n}{1-u^2/n}\Big\}\Big)\,du \\
&\le\...
9
votes
Estimates for certain double-Kloosterman sums
I can do it for $q$ prime, and thus $q$ squarefree, by a long but elementary manipulation followed by the Weil bound. Until the end, the elementary manipulations will work for $q$ arbitrary.
Taking ...
9
votes
Accepted
Cancellation in a very rapidly oscillating exponential sum
I doubt that one is able to get as far as $T = \exp(\log^{2-\varepsilon} x)$ with Weyl differencing. Standard Weyl differencing arguments, such as that in Theorem 8.4 of
Iwaniec, Henryk; Kowalski, ...
9
votes
Accepted
Does there exist some irrational $x,\alpha$ so that this Weyl sum is $o(\sqrt N)$?
The estimate
$$ S_N(x) = S_N(x,\alpha) = o(N^{1/2}) \tag {1}$$
cannot hold when $x$ is irrational.
Heuristically, the reason comes from the (approximate) modularity properties of $S_N(x,\alpha)$; this ...
8
votes
Accepted
The first case of the strong Littlewood conjecture
The following human-verifiable proof is in collaboration* with Fedja.
Lemma $1$: We have the following for $0 \le x \le 3$: $$\frac{6204}{6750}x^2-\frac{8429}{60750}x^4+\frac{4475}{546750}x^6\le x.$$
...
8
votes
Accepted
Is the Lebesgue measure of the $x$ so that this exponential sum is $o(\sqrt{N})$ positive?
Ok, apologies if this is overkill, but this paper shows that for almost every (irrational) $\alpha \in \mathbb{R}$, only a measure $0$ set of $x \in \mathbb{R}$ satisfy $S_{N,\alpha}(x) = o(\sqrt{N})$ ...
7
votes
Sums of twisted products of Kloosterman Sums
Your sum is $$\sum_{i=0}^2 (-1)^i \operatorname{tr}(\operatorname{Frob}_p, H^i_c( \mathbb A^1_{\overline{\mathbb F}_p}, \mathcal{K}\ell_2 (ab) \otimes \mathcal{K}\ell_2 (a(b+h)) \otimes \mathcal L_\...
7
votes
Hypotheses for exponent pairs
See Montgomery's book, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, specifically the examples on p.55. He considers variants of your sum, namely $\sum_{n \leq N}...
7
votes
Accepted
Moments of certain exponential sum
Yes. Without loss of generality, $X\geq 2$ is an integer. Then we have explicitly
$$ v(\beta)=e\left(\frac{(X+1)\beta}{2}\right)\frac{\sin(\pi X\beta)}{\sin(\pi\beta)},\qquad\beta\not\in\mathbb{Z}.$$
...
7
votes
Accepted
Why are exponential sums so bad at solving this very easy problem?
The basic example of exponential sums in Number Theory is to count solutions to an equation such as $f(x_1,\cdots,x_n)\equiv 0$ $\mod p$ where $p$ is prime this is because $p$-adic solutions are a ...
7
votes
Accepted
Bound for sum of multiplicative character calculated over multivariate polynomial
Yes, there are several known bounds. The following statement, due to Katz, has quite strict conditions on $f$, but gives a very strong result. It is perhaps the simplest statement that gives such a ...
7
votes
On the upper-bound for a type of quintuple Kloosterman sums
The Newton polyhedron method for your sum is as follows:
We first replace the quadratic character with an additive character and a simpler quadratic character using a Gauss sum, then introduce a ...
7
votes
On the estimate for the mixed 3-dimensional hyper-Kloosterman sum
This sum can be handled by elementary change of variables.
Let $a = \overline{z} x$, $b=y$, and $c =\overline{xy}$. Then $x = \overline{bc}$, $y=b$, and $z=\overline{abc}$, so $(x,y,z) \to (a,b,c)$ ...
6
votes
Accepted
estimate for a sum of products of Weil's sum
Here is a different approach than Denis Chaperon de Lauzières's.
Opening everything and using orthogonality of characters to remove the $x$ variable, we see that your sum is $1/p$ times $$\sum_{z_1,...
6
votes
Accepted
An Exponential Sum Restricted to Primes
No, this is definitely not true. First of all, there's no way we should expect better than square-root cancellation in an exponential sum without an incredible amount of structure, which the primes do ...
6
votes
Summing the infinite series $\sum_{k=0}^{\infty} \frac{x^k}{(k!)^2}$
The first way I try to solve questions like this is to "ask Maple" (or Mathematica). If you have access to, say, Maple, then you can type
"sum(x^k/k!^2, k=0..infinity)"
and it will report BesselI(0,...
6
votes
Accepted
Integral over an exponential sum with squares
The integral is equal to exactly one - just expand out the square and use orthogonality to get
$$ I = \sum_{0\leq n,m<p} \int_0^1 e((n^2+m^2)t) \mathrm{d} t = \sum_{0\leq n,m<p}1_{n^2+m^2=0}=1.$...
6
votes
Accepted
Can the following sum be counted or expressed in terms of special functions?
Probably the answer is negative. Your series is a restriction of the analytic function in two complex variables:
$$F(\zeta,q)=\sum_{n=0}^\infty\frac{q^{n^2}}{n!}\zeta^n,\quad |q|\leq1,$$
obtained by ...
Only top scored, non community-wiki answers of a minimum length are eligible
Related Tags
exponential-sums × 185nt.number-theory × 126
analytic-number-theory × 94
sequences-and-series × 16
fourier-analysis × 16
reference-request × 14
ag.algebraic-geometry × 12
co.combinatorics × 10
harmonic-analysis × 10
finite-fields × 10
real-analysis × 7
polynomials × 6
inequalities × 6
cohomology × 6
additive-combinatorics × 6
ca.classical-analysis-and-odes × 5
prime-numbers × 5
diophantine-equations × 5
equidistribution × 5
pr.probability × 4
cv.complex-variables × 4
algebraic-number-theory × 4
arithmetic-geometry × 4
modular-forms × 4
exponential-polynomials × 4