All Questions
Tagged with analytic-number-theory convergence-divergence
80
questions
1
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0
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92
views
Inequalities for the Dirichlet eta function at non-trivial Riemann zeta zeros.
I am interested in these inequalities for sufficiently large $n$:
$$\Large \left(\Re\left(\sum _{k=1}^n (-1)^{k+1} x^{\log (k) c}\right)\right)^2 \leq \left( \Re\left(x^{\log \left(n+\frac{1}{2}\right)...
- 7,448
0
votes
0
answers
82
views
Clarification about argument why $\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}$ diverge
I would like to clarify some aspects in this answer by Noam D. Elkies proving divergence of
$$\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}.$$
Firstly, do I understand the general strategy ...
- 7,499
1
vote
2
answers
53
views
Is there an analytic continuation of the Legendre Chi function $\chi_2(z)$ for $z > 1$?
The Legendre Chi function $\chi_2(z)$ is define as
$$
\chi_2(z) = \sum_{k=0}^{\infty}\frac{z^{2k+1}}{(2k+1)^2}
$$
for $-1 \le z \le 1$. But $z > 1$ the series diverges. For real value of $z$ is ...
- 20.8k
1
vote
0
answers
42
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Convergence of Riemann zeta function [duplicate]
I am wondering whether the series $$\zeta(s) = \sum_{n=1}^\infty n^{-s}$$ converges for $s$ with $\mathsf{Re}(s)=1$ and $\mathsf{Im}(s) \neq 0$. Note that I use the representation of $\zeta$ as an ...
- 1,803
1
vote
0
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75
views
A question about Lemma 15.1 (Landau’s theorem for integrals) in Montgomery-Vaughan’s book
Lemma 15.1 in Montgomery-Vaughan’s analytic number theory book is Landau’s theorem for integrals. My question is, why is it necessary to have $A(x)$ bounded on every interval $[1,X]$? Doesn’t the ...
- 405
0
votes
1
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61
views
A converge problem seems to related to Tauber theorem
ps: Due to my poor English, I might describe my thought roughly.
Suppose $\{a_n\} (n \ge 0)$ is a sequence consisting of non-negative numbers,
and $\{a_n\}$ satisfies that forall $x > 0$, $f(x) = \...
2
votes
1
answer
46
views
$\lim_{s\to1}\sum_{p\equiv a \text{ mod } q} p^{-s} = \sum_{p\equiv a \text{ mod } q} p^{-1}$?
In the context of the Dirichlet Prime Theorem, we come to the equation
$$\sum_{p\equiv a \text{ mod } q} p^{-s} = \varphi(q)^{-1}\sum_{p \not\mid q}p^{-s} + \varphi(q)^{-1}\sum_{\chi\not=\chi_0}\bar{\...
- 901
4
votes
1
answer
109
views
Abscissa of convergence of a Dirichlet series with bounded coefficients and analytic continuation [closed]
If a Dirichlet series has coefficients +1 and -1 and an analytic continuation without poles (or zeros) to the right of Re(s) = 1/2, what can we say about it's abscissa of convergence?
Is it always at ...
- 41
3
votes
1
answer
191
views
If $\sum_{n=1}^\infty\frac1{na_n}$ converges, then $\sum_{n=1}^\infty\frac1{na_{b_n}}$ is also convergent
I'm trying to determine whether the following statement is true or false:
Let $b_n$ be the largest prime factor of the positive integer $n$, and $\{a_n\}$ be a increasing positive sequence such that $...
- 13.7k
4
votes
1
answer
179
views
One series converges iff the other converges
In this post Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges it is said that this series converges $$\sum_{1<n\leq N}\frac{a_{n}}{\log\left(n\right)}=\!\...
- 49
0
votes
0
answers
136
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Show that $\sum a(n)/\log n$ converges if and only if $\sum a(p)$ converges. [duplicate]
I am studying analytic number theory at the time and I came across with this. I was wondering if there is a somehow smart way to prove the following without using prime number theorem(but probably use ...
- 49
0
votes
1
answer
82
views
Is the convergence of these two series equivalent? (They come from Khinchin's theorem and the Duffin-Schaeffer conjecture.)
I am trying to wrap my head around two theorems of Diophantine approximation: Khinchin's theorem and the Duffin and Schaeffer conjecture. To the best of my understanding, here is what they say:
...
- 392
2
votes
1
answer
160
views
If a subset of the primes has converging reciprocal sums, does it have relative density zero?
If a subset $A\subset\mathbb{N}$ of the natural numbers has positive lower density, i.e. there is an $\epsilon>0$ and a sequence $\{N_k\}$ of natural numbers s.t. $$\liminf_{k\to\infty}\frac{\lvert ...
0
votes
0
answers
34
views
Justify convergence and differentiation of $\frac{\zeta'}{\zeta}(s)$
Using the Euler-Product of the $\zeta$ function and taking the $\log$ of it, it is easy to derive the equation
$$\log\zeta(s) =\sum_p\sum_{k=1}^\infty\frac{p^{-ks}}{k}$$
However, what my textbook does ...
- 901
0
votes
0
answers
266
views
Proof Riemann Zeta Series based on $\eta(s)$ has only one pole.
This proof is my understanding of a very interesting comment by @leoli1 on my previous related question about the following extended Riemann Zeta function which converges for $\sigma>0$ where $s=\...
- 3,325