All Questions
Tagged with analytic-number-theory sequences-and-series
293
questions
2
votes
1
answer
55
views
General form of Jacobi Theta Transformation $\sum_{n \in \mathbb{Z}} e^{n^2 \pi / x} = \sqrt{x} \sum_{n \in \mathbb{Z}} e^{n^2 \pi x} $
I was looking into the functional equation of $\zeta(s)$ and at one point the proof uses the Jacobi Theta Transformation:
$$\sum_{n \in \mathbb{Z}} e^{n^2 \pi / x} = \sqrt{x} \sum_{n \in \mathbb{Z}} ...
- 730
2
votes
2
answers
216
views
Is $\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$ not convergent?
I need to calculate the limit (if it exist) $$\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$$ where $\{x\}$ denotes the fractional part of $x$, $n!$ is the factorial of $n$...
- 928
1
vote
1
answer
95
views
Non negativity involving sequences
Define for $n\in\mathbb{N}$ $$a_n=\left[n\sum_{k=1}^{n}\frac{1}{k^5}\right]$$ where $[x]$ denotes the greatest integer $\leq x$.
Prove that $$(n+1)a_n-n a_{n+1}+1\geq 0 \ \ \forall n\geq 1$$
$$(n+1)...
- 928
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
2
votes
1
answer
70
views
Differently defined Cesàro summability implies Abel summability
I am trying to solve the Exercise 10 of Section 5.2 of the book `Multiplicative Number Theory I. Classical Theory' by Montgomery & Vaughan. In the exercise, they define the Cesàro summability of ...
- 1,145
1
vote
0
answers
42
views
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
0
votes
0
answers
40
views
Two slightly different polynomial expansions for $\Xi(0)$. Could a connection between these two be derived?
After experimenting with polynomial expansions for the Riemann $\Xi(t)=\xi\left(\frac12+it\right)$-function, I landed on these two equations:
with $M$ the KummerM confluent hypergeometric funcion and ...
- 3,191
3
votes
0
answers
116
views
Visual proof of $\sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) = 1 $
Background
Let $\zeta(\cdot)$ be the Riemann zeta function. I'm looking for a visual proof of the infinite series identity $$\sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) = 1. \tag{1}\label{1}$$
This ...
- 7,118
9
votes
2
answers
319
views
An accurate, rapidly converging estimate of $\sum_{j = 1}^{\infty}\frac 1 {j^2}$ (the Basel Problem) using only elementary calculus
Although the Basel Problem $$\zeta(2) = \sum_{j = 1}^{\infty}\frac 1 {j^2}$$ took a hundred years to solve analytically, using elementary calculus we can easily get strikingly accurate bounds: $$\zeta(...
- 4,450
1
vote
1
answer
101
views
An infinite series of powers of fractions.
The series at hand is given by,
$$\sum_{k=1}^{\infty} \left(\frac{2k+1}{k (k+1)}\right)^s$$
I think it converges for $s>1$, but I have not been able to derive a general expression for this series.
...
- 33
1
vote
2
answers
106
views
What is the limit of $\frac{\Omega(n!)}{n}$ as $n \to \infty$?
Let $p_i$ be the $i$th prime number, and $\Omega(n) =$ the total number of prime factors of $n$ counting multiplicities. We know that $\Omega(n)$ is completely additive, so:
$$
\frac{\Omega(n!)}{n} ...
2
votes
1
answer
132
views
Analytic Number Theory [closed]
I really need help with two problems that I can't solve.
I have the following function: $$R(N) = \sum_{n=1}^N\exp(2\pi i\sqrt{n})$$
I was able to show that $R(n)=O(\sqrt{n})$, but I am also being ...
- 111
1
vote
0
answers
76
views
Dirichlet generating function that generates itself
Background
There are a few questions on this venue on functions that generate themselves:
In this question, a function is sought that is its own ordinary generating function. An answer is obtained ...
- 7,118
4
votes
0
answers
74
views
Can we extend the Divisor Function $\sigma_s$ to $\mathbb{Q}$ by extending Ramanujan Sums $c_n$ to $\mathbb{Q}$?
It can be shown that the divisor function $\sigma_s(k)=\sum_{d\vert k} d^s$ defined for $k\in\mathbb{Z}^+$ can be expressed as a Dirichlet series with the Ramanujan sums $c_n(k):=\sum\limits_{m\in(\...
- 1,810
0
votes
0
answers
55
views
Swapping finite product with infinite sum
I have from my analytic number theory notes that states $$\prod_{p\le P} \sum_{k=0}^\infty \frac{1}{p^{ks}} = \sum_{n=1}^\infty \frac{c_P(n)}{n^s}$$ for primes $p$ and $Re(s)>1$. $c_P(n)$ is $1$ ...
- 5,601