All Questions
Tagged with analytic-number-theory inequalities
47 questions
1
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0
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100
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Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$
$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$.
I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
- 178
5
votes
2
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355
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Can one show that $(-1)^{n-1} {(1/\zeta)}^{(n)}(x) >0$ for all real $x>1$?
Is it true that $(-1)^{n-1} {(1/\zeta)}^{(n)}(x) >0$ for all real $x>1$ ?
Or in other words can you show that the higher order derivatives of the reciprocal of the Riemann zeta function ...
- 178
0
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2
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364
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Can one show $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the ...
- 178
6
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1
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568
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Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{\zeta'(x)}{\zeta^2(x)}\right|\leq \frac{1}{x-\frac{1}{2}}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the inequality and got this inequality ...
- 178
2
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0
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159
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Upper bound of a product of sines
Consider the function
$$ f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$
I wonder whether it is possible to compute some nontrivial upper ...
- 826
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357
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Bounding a sum of reciprocals of square-free integers
(Cross-posted from MSE, as the question did not get any clear answer)
Fix positive integers $k$ and $n$. Let $N_1,\dots,N_r$ be all the integers less than or equal to $n$ that are squarefree and have ...
- 189
4
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1
answer
353
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Inequalities involving binary representation of integers
Let $N\geq 1$ be a positive integer and assume that $N=2^{n_1}+2^{n_2}+\cdots+2^{n_{p}}$, $n_{1}>n_{2}>\cdots>n_{p}\geq 0$, is the binary representation of $N$. I believe that the following ...
- 41
10
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2
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849
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Schur's proof of Hilbert's inequality: streamlining?
TL;DR: Is there a way to make Schur's (elegant) proof of Hilbert's inequality feel like
less of a trick/miracle?
Longer version: Let me go quickly over Schur's proof to show what I mean. Actually, let ...
- 20.2k
10
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2
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282
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Non-trivial upper bound for a sum related to $p^{-1}z \pmod q$ and $q^{-1}z \pmod p$
Let $p, q$ be two distinct prime number. I'm trying to provide a non-trivial upper bound for the sum
$$S(p, q) = \sum_{1 \leq x < p} \sum_{1 \leq y < q} \frac{1}{\|x / p\| \, \|y / q\| \, \|x/p +...
- 65
1
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0
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305
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About inequalities that involve the sum of divisors, the Euler's totient and the aliquot part $\sigma(n)-n$
In this post, for integers $n\geq 1$, I denote the sum of divisors $\sum_{1\leq d\mid n}d$ as $\sigma(n)$ and the Euler's totient function as $\varphi(n)$. It's easy to check* that if we assume that $...
3
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151
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On an inequality for the arithmetic function counting the number of primes $\lfloor n^c\rfloor$ in the spirit of Ramanujan's prime counting inequality
In page 3 of [1] (please see if you need it the book by Berndt) Axler refers an inequality that involves the prime-counting function $\pi(x)$ and that was deduced by Ramanujan. I'm curious to know if ...
2
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0
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110
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On variations of a claim due to Kaneko in terms of Lehmer means
This post is cross posted from Mathematics Stack Exchange, due that there was a mistake from my part (see the excellent partial answer and my thread of edits of my question on MSE) this post on ...
0
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1
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230
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Inequality in Iwaniec-Kowalski
I am reading about Dirichlet polynomials in the book Analytic Number Theory by the said authors. Can anyone justify the following inequality? Assume that $a(n),b(m)$ are sequences of non-negative ...
- 3,062
5
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1
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391
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Proving a specific case of Robin's Inequality
Edit: It turns out that this is equivalent to the RH which gives the idea that this might a a little difficult to show. As such we could consider an even simpler case in which the number $n$ is ...
- 315
0
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0
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106
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Variants of Nicholson's inequalities for prime numbers, involving the Lambert $W$ function
The purpose of this post is ask about two closely related/inspired conjectures from inequalities due to Nicholson (see [1]) and Visser [2]. If my reasonings are right should be stronger versions of ...
0
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0
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219
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On an inequality involving the Lambert $W$ function and the sum of divisors function
Let $W(n)$ be the principal/main branch of the Lambert $W$ function (this is the Wikipedia related to this special function). I was inspired in Robin equivalence to the Riemann hypothesis (see [1]) ...
2
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1
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146
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On an attempt to create interesting variants of Firoozbakht's conjecture, evoking combinations of different generalized means
The idea of this post arises when I've considered simple variants of the known as Firoozbakht's conjecture (see this corresponding Wikipedia or [1]), and comparisons by trial and error with means (if ...
0
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0
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158
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Similar problems for the Dedekind psi function than those that are in the literature for the Euler's totient function: reference request or proposals
I know the linked articles from Wikipedia for the Euler's totient function and the Dedekind psi function. I know that are in the literature famous, interesting and difficult problems involving the ...
1
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0
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315
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From an inequality for the Euler's totient function to a combination of Firoozbakht's conjecture and Nicolas' criterion for the Riemann hypothesis
In this post we ask about the veracity of an inequality deduced from a combination of Firoozbakht's conjecture (see [1] or [2]) and Nicolas' criterion for the Riemann hypothesis (see for instance [3])....
-3
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2
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261
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The inequality $\Pi (1-\frac{1}{a_i})^{x_i} \le \Pi (1-\frac{1}{b_j})^{y_j} $ hold? [closed]
Question: Are the properties as follows holds?
Version 1: the answer by Bjørn Kjos-Hanssen
Let $P$ be a positive integers. We written: $P=$ $a_1^{x_1}a_2^{x_2}...a_n^{x_n}$ $=b_1^{y_1}b_2^{y_2}......
- 1,776
4
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2
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472
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Sharp estimates for Meissel-Mertens constant
I wondered if it is possible to get a similar inequality like $(1.1)$ of Michael D. Hirschhorn, Approximating Euler's Constant, The Fibonacci Quarterly, Volume 49, Number 3 (August 2011) for the ...
-3
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1
answer
281
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A Bonse's inequality for semiprimes, with a good mathematical content
A semiprime $s$ is a positive integer that is the product of two prime numbers, see Semiprine the encyclopedia Wikipedia. A well-known inequality, with applications, that involves prime numbers is the ...
5
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1
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268
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Is it a known property of positive integers $n> 2 $ that one must have $n < \mathrm{rad}(n(n-1)(n-2))$?
Let $P(n)$ be the statement that
$$n < \mathrm{rad}(n(n-1)(n-2)),$$
where $\mathrm{rad}$ is the radical of an integer, that is defined as
$$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ ...
- 1,776
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0
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83
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Is it possible to get a conjecture similar to Mandl's conjecture for a different arithmetic function of number theory, mainly related to primes?
I'm curious to know if are in the literature analogous conjectures to the conjecture due to Mandl, I ask about these analogous conjectures for different sequences playing an important role in number ...
1
vote
1
answer
288
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Bounding Coefficients of Dirichlet Series
Consider the exponentiated Riemann-Zeta function $\zeta(s)^p$. If it is represented as
$$\zeta(s)^p = \sum_{n=1}^\infty\frac{a_n}{n^s}$$
Is there any upper bound we can put on $|a_n|$ in terms of ...
- 1,129
1
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1
answer
195
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A simple inequality that arises from the exact form for the prime-counting function and the second Hardy–Littlewood conjecture
The germ of this post arises when I was trying to think what terms of the so-called exact form of the prime-counting function satisfy an inequality of the form $\text{term}(x+y)\leq \text{term}(x)+\...
15
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5
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2k
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$|L'(1,\chi)/L(1,\chi)|$
Let $\chi$ be a primitive Dirichlet character $\mod q$, $q>1$. Is there a neat, simple way to give a good bound on $L'(1,\chi)/L(1,\chi)$?
Assuming no zeroes $s=\sigma+it$ of $L(s,\chi)$ satisfy $\...
- 20.2k
2
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1
answer
127
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Effectiveness of the implied constant in an inequality
A result of Heath-Brown states:
For $a_1,...,a_n$ be arbitrary complex numbers,
$$\sideset{}{^*}\sum_{m\le M} \left|\sideset{}{^*}\sum_{n\le M}a_n\left(\frac{n}{m}\right)\right|^{2} \ll_{\epsilon}(MN)^...
- 29
13
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2
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730
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Show that there exist $k\in\{1,2,\cdots,n\}$ such that $\frac{1}{n}\sum_{i=1}^{n}\left(\{kx_{i}\}-\frac{1}{2}\right)^2>\frac{1}{12}-\frac{1}{6n}$
The following question has post mathsatck :Nice problem, perhaps from a question of expectation.The problem seemed so elementary, but I had tried to solve it for a long time and eventually failed.I 'd ...
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1
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1
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141
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Compare $\operatorname{rad}(an+b)$ and $\varphi(cn+d)$ in a simple and interesting inequality, for some choice of integers $a,b,c$ and $d$
We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(1)=1$. You can see this ...
user75829
0
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1
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125
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Polyhedral conditions for quadratic inequalities in fixed dimension
Denote $\mathcal T$ be set of $(T_1,T_2,T_3,T_4)\in\mathbb Z^4$ that satisfy
$$0<T_1,T_2,T_3,T_4$$
conditions?
Define the level set $$M_{\gamma}(Q,\mathcal T)=\{(T_1,T_2,T_3,T_4)\in\mathcal T:Q(...
- 13.9k
6
votes
1
answer
287
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Number of solutions for the inequality with square roots
Let $M$ be some large real number and $\delta>0$. I would like to estimate the number of solutions for the inequality
$$|\sqrt{n_1}+\sqrt{n_2}-\sqrt{n_3}-\sqrt{n_4}|<\delta\sqrt{M},$$
where $...
- 7,193
10
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1
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1k
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The supremum value of $\int f(t) \log{\frac{1}{|t|}} \, dt$ for normalized Fourier pairs non-negative outside of $[-1,1]$
Observe that for any Schwartz function $f \in \mathcal{S}(\mathbb{R})$ having
$$
f(0) = \widehat{f}(0) = 1
$$
and
$$
f, \widehat{f} \geq 0 \quad \textrm{outside of} \quad [-1,1],
$$
the following ...
- 13.8k
2
votes
2
answers
1k
views
Estimates for Sum of Prime Factors and Number of Prime Factors
Given a positive integer $n$, I've workout out a formula which involves the expression "sum of distinct primes dividing n" minus "number of distinct prime factors of n."
Are there any known ...
- 329
1
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2
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209
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What is the upper bound for $\int \limits_{2}^{x} \frac{e^{-0.3\sqrt{\ln(t)}}}{\ln^2(t)} dt$?
For start, is $\int \limits_{2}^{x} \frac{e^{-0.3\sqrt{\ln(t)}}}{\ln^2(t)} dt \leq x\ln(\ln(x)) \frac{e^{-0.3\sqrt{\ln(x)}}}{\ln^2(x)}$ ?
If the above is true, what is a better bound for the integral ...
- 185
5
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1
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436
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Even sharper upper bound for prime product?
In Dusart papers he proves that $\prod \limits_{p \leq x} \frac{p_i}{p_i-1} \leq e^\gamma \ln (x) \left(1+\frac{0.2}{\ln ^2 (x)} \right)$ for large numbers.
What I am asking is could we make the ...
user95470
0
votes
1
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437
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Mertens' 3rd theorem, upper bound
Is it true that
$$\prod_{p\le x}\frac p{p-1}\le e^\gamma\ln x\left(1-\frac{0{.}011}{\ln x}+\frac{0.2}{(\ln x)^2}\right)$$
for all $x>25\,000$, where the product is over prime $p$?
user95470
-1
votes
1
answer
199
views
Find the values of $n$ that satisfy this inequality involving a product over prime numbers [closed]
Inequality
What values of $n$ satisfy the following inequality?
$$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$
$p$ are prime numbers and the notation $p_i$ indicates the $i$th ...
- 161
4
votes
1
answer
419
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The values of $n$ which satisfy an inequality about prime numbers
For which values of $n$ does the following inequality hold for?
$$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-1}{p_i}\right)$$
$p$ are prime numbers and the notation $p_i$ indicates the $i$-th ...
- 161
2
votes
1
answer
222
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trigonometric sum and inequalities
let $x\in\mathbb{R}-\mathbb{Z}$ and $e(x)=e^{2\pi ix}$. If we have this sum $$\left|\overset{q}{\underset{h=1}{\sum}^{*}}e\left(h\, x\right)\underset{\underset{p\equiv h\,\textrm{mod}\, q}{p\leq N}}{\...
- 45
4
votes
2
answers
366
views
Inequality for an integral involving $ \exp $, $ \sin $ and $ \cos $
Let $ t > 0 $ and $ k \in \{ 0,1,2,\ldots \} $. Does the following inequality hold?
$$
\int_{k + 1/2}^{k + 3/2}
\frac{x \sin(2 \pi x)}{1 + 2 e^{2 \pi t} \cos(2 \pi x) + e^{4 \pi t}}
...
3
votes
1
answer
921
views
Number of solutions in a sum of squares Diophantine equation
Let $n$ be an integer. I'm interested in upper bounding the number of integer solutions of
\begin{equation*}
x_1^2+x_2^2+\ldots+x_p^2=y_1^2+y_2^2+\ldots+y_p^2,
\end{equation*}
where for all $i\in\{1,...
- 859
10
votes
1
answer
1k
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Can the Brun-Titchmarsh theorem be improved when the modulus is smooth?
For $q,a$ relatively prime, let $\pi(x,q,a)$ denote the number of primes less than $x$ which are congruent to $a$ modulo $q$. The Brun-Titchmarsh theorem states that $$\pi(x,q,a)\leq \frac{(2+o(1))x}{\...
- 11.4k
3
votes
0
answers
947
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Inequalities in paper by Jean Bourgain
The question refers to the following paper by Jean Bourgain: http://arxiv.org/abs/math-ph/0011053
Specifically, I can't derive the following inequality in (1.20):
\begin{equation}
\left|\sum_{|k|\...
11
votes
1
answer
1k
views
The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$
The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
a\...
- 11.4k
3
votes
4
answers
678
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Equidistribution Theorem: distance between solutions
Can please someone help me with the following problem.
Say we have a sequence $nx \; \mathrm{mod} \; 1$, where $n$ is a whole number and $x$ is irrational.
Now I need to solve the inequality
$nx \; \...
- 33
3
votes
2
answers
597
views
lower bound for $\Re\zeta(1+it)$
Hi
is there any lower bound for $\Re\zeta(1+it)$.
I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$.
If it is true, is there any reference to prove it.
thanks
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