Questions tagged [lower-bounds]

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Quantitative lower bounds related to Zhang's theorem on bounded gaps

Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_{...
Eric Naslund's user avatar
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24 votes
4 answers
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Super-linear time complexity lower bounds for any natural problem in NP?

Do we know any problem in NP which has a super-linear time complexity lower bound? Ideally, we would like to show that 3SAT has super-polynomial lower bounds, but I guess we're far away from that. I'd ...
Rune's user avatar
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8 votes
2 answers
704 views

Lower bound on the entries of the Perron vector

Let $A$ be a matrix that satisfies all the conditions of Perron- Frobenius theorem. From the theorem it is known that the entries of the eigenvector corresponding to the largest eigenvalue will be ...
biryani's user avatar
  • 183
8 votes
2 answers
913 views

Lower bound on exponential sums

Let $k\geq 2$. Consider the following norm of exponenetial sum: $$ I(N,p,k)=\int_0^1\int_0^1 \left|\sum_{n=0}^N e^{2\pi i (n x+n^k y)}\right|^p dxdy. $$ Bourgain mentioned on Page 118 of https://...
Thomas Yang's user avatar
8 votes
1 answer
581 views

lower and upper bound for $\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k$?

Are there known any lower and upper bounds for $$ \sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k, $$ where $\Omega(n)$ is the number of prime factors counting multiplicities of $n$? Or at least is it known ...
asad's user avatar
  • 841
6 votes
2 answers
722 views

bound for zeros of a polynomial with bounded integer coefficients

Let $f$ be a monic polynomial with bounded integer coefficients and such that all zeros are (in absolute value) greater than $1$. How close can the zeros of $f$ reach $1$ (in absolute value)? More ...
Daniel Krenn's user avatar
5 votes
4 answers
816 views

Limit of a sum with binomial coefficients

Let $$A_k = \frac{\sum_{i=1}^ki{2k-i-1 \choose i-1}{i-1 \choose k-i}}{k{2k-1\choose k}}$$ $$B_k = \frac{\sum_{i=1}^ki{2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}$$ $$C_k = \frac{\sum_{i=1}^k(...
macat's user avatar
  • 145
4 votes
1 answer
157 views

Existence of copula bound pointwise strictly smaller than the Fréchet-Hoeffding upper bound

Consider bivariate copulas $C_1$ and $C_2$ with $\max\{C_1(u,v), C_2(u,v)\}< M_2(u,v)$ for all $u,v \in(0,1)$, where $M_2(u,v) := \min\{u,v\}$ is the Fréchet-Hoeffding upper bound. Is there a ...
Corram's user avatar
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3 votes
1 answer
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Lower bounds on the error term of the prime number theorem

Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t. $$f(x)\ll |\psi(x) - x|$$ where $\psi$ is the Chebyshev function.
Mayank Pandey's user avatar
3 votes
0 answers
661 views

Counting / characterizing the isolated points of a particular algebraic variety

I'm not a professional geometer / topologist, so please thanks for your patience :) Setup The following questions are the first in a series of steps I'm undertaking in an attempt to break down a ...
dohmatob's user avatar
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2 votes
0 answers
275 views

Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues

In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ ...
dtn's user avatar
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