5
votes
1answer
54 views

Sum of a certain series related to the primes

It is well known that $$\sum_{n > 0}\frac{1}{n}$$ diverges, but $$\sum_{n > 0}\frac{1}{n^2} = \frac{\pi^2}{6}$$ converges. Similarly, $$\sum_{p}\frac{1}{p}$$ diverges, but $$\sum_{p} ...
3
votes
0answers
112 views

Ratio of maximal to minimal jump in the set of angle multiples (corrected)

(This is the corrected version of the question I asked here: Ratio of maximal to minimal jump in the set of angle multiples.) Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times ...
13
votes
3answers
521 views

A closed form for the sum $\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$

How can I find a closed form for the following sum? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
2
votes
1answer
79 views

Absolute convergence of Euler products and infinite series

We know that given a multiplicative function $f$ for which the series $\sum_{n=1}^\infty f(n)$ converges absolutely then so does the Euler product $\prod_{p}\sum_{k=0}^\infty f(p^k)$, but does the ...
4
votes
1answer
143 views

Prime harmonic series

We have following identity: ($p$ is a prime number) $$\left(1+\frac{1}{p}\right)\sum_{k=0}^n\frac{1}{p^{2k}}=\sum_{k=0}^{2n+1}\frac{1}{p^k}$$ Now, How to derive the following inequality from the above ...
6
votes
1answer
186 views

Newman's “Natural proof”(Analytic) of Prime Number Theorem (1980)

I am trying to understand this short proof by newmann. I faced some problems while grasping this very proof. Please help me out. 1 . I am not clear, why in step (1)'s proof he says that from unique ...
3
votes
1answer
117 views

Convergence of the Zeta and Phi functions

I want to show that the following functions (in the picture) are absolutely and locally uniformly convergent if real part of complex number $s$ is bigger than 1. Absolute part for zeta function is ...
1
vote
1answer
64 views

$|A(n)|<B$, $\lim_{s \to 0^{+}}\sum_{n=1}^{\infty}a_{n}n^{-s}=a$ imply $\lim_{x \to \infty}\sum_{n \leq x} a_{n}(1-\log{n}/\log{x})=a$

Suppose that $|A(n)|<B$ and $\lim_{s \to 0^{+}}\sum_{n=1}^{\infty}a_{n}n^{-s}=a$ where $A(x)=\sum_{n \leq x}a_{n}$. Then $$\lim_{x \to \infty}\sum_{n \leq x} a_{n}(1-\log{n}/\log{x})=a$$ What I ...
6
votes
5answers
211 views

$(1+\frac{1}{n\log n})^n-1=O(\frac{1}{n})$.

When I solved a problem, I could solve it if I assumed that $(1+\frac{1}{n\log n})^n-1=O(\frac{1}{n})$ I tried to prove it, but I failed. Actually, I don't convince if it is true. Is it correct? If ...
1
vote
3answers
74 views

Questions about assigning a probability to a randomly chosen large integer $n$ being prime

I heard this question a few days ago, so reciting from memory: If I were to randomly choose an arbitrarily large positive integer $n$, could I write a function that determines the likelihood of it ...
5
votes
3answers
224 views

From $\sum_p \frac{\log p}{p^s} = \frac{1}{s-1} + O(1)$ conclude that $\sum_p \frac{1}{p^s} = \log \frac{1}{s-1} + O(1)$

I'm reading a book on analytic number theory. It asks me to prove: $$ \sum_p \frac{\log p}{p^s} = \frac{1}{s-1} + O(1) \tag{A}$$ and conclude, via integration, that: $$ \sum_p \frac{1}{p^s} ...
0
votes
1answer
64 views

Question regarding the function $R_X(t)=\frac{1}{\pi} \sum_{p\leq x} \frac{\sin(t\log p)}{\sqrt{p}}$

I want to show that the expected value $\mathbb{E}_{\omega ,T}(R_x(t)^{2k})$ behaves asymptotically as: $$\frac{(2k)!}{k!\cdot 2^k} \left(\frac{\log(\log T)}{2\pi^2}\right)^k$$ for $T^\epsilon < ...
4
votes
1answer
281 views

Evaluating a limit of the truncated exponential series motivated by the prime number theorem for $k$ distinct prime factors.

If $\pi_k(n)$ is the cardinality of numbers with k factors (repetitions included) less than or equal n, the generalized Prime Number Theorem is: $$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln \ln ...
9
votes
1answer
190 views

Fourier Series involving the Jacobi Symbol

We know that the Fourier Series $$s(x)=\sum_{k\neq0}\frac{1}{k}\exp\left(2\pi ik x\right)$$ corresponds to the sawtooth function, $s(x)=\left\{x\right\} -\frac{1}{2}$. Suppose that ...
7
votes
2answers
179 views

On a double sum involving prime numbers

$$\sum_{i,j=1}^{\infty}\left[\frac{x}{p_ip_j}\right]=x\sum_{p_ip_j\leq x}\frac{1}{p_ip_j}+O(x),p_i< p_j$$ where $p_i$ is the $i$th prime, and "[ ]" represents the largest integer not ...
8
votes
2answers
468 views

Equivalence to the prime number theorem

I was just reading this question and answer: How will this equation imply PNT and it raised a whole new question: Given that $\sum_{n\le x} \Lambda(n)=x+o(x)$, prove that $$\sum_{n\le x} ...
8
votes
1answer
212 views

Analytic number theory primer — sequences and series

For a book like Titchmarsh, or Iwaniec and Kowalski, it seems a thorough knowledge of asymptotics is a prerequisite. What are good books for training oneself in such manipulation of asymptotics, ...
5
votes
1answer
132 views

An estimate of a series

Suppose $s$ is not an integer, let $\lambda(s)=\min_{n≥0}|s+n|$. Show that $\sum\limits_{n=1}^{\infty}(\frac{1}{n+s}-\frac{1}{n})\ll\frac{1}{\lambda(s)}+\log(|s|+2)$.
6
votes
2answers
613 views

Asymptotic formula for $\sum_{n\leq x}\mu(n)[x/n]^2$ and the Totient summatory function $\sum_{n\leq x} \phi(n)$

I would like to show (for $x \ge 2$) that $$\sum_{n \le x}\mu(n)\left[\frac{x}{n}\right]^2 = \frac{x^2}{\zeta(2)} + O(x \log(x)).$$ I already have the identity $$\sum_{n \le ...
7
votes
0answers
266 views

How can we prove a simple case of the High Indices Theorem?

Let $(a_n)$ be a sequence of real numbers such that $$f(x) = \sum_{n=1}^{\infty} a_n x^{2^n}$$ converges for $|x| < 1$ and $f(x)$ converges to $a$ as $x \to 1^{-}$. Then I have to prove that $\sum ...
5
votes
1answer
184 views

Abelian theorem regarding Riesz summability

This is my first time to post something here. If there is anything wrong, please inform me... Anyway, here is my question: Let $k$ be a nonnegative integer. We say a sequence $(a_n)$ is $(R, ...
21
votes
3answers
391 views

Computing the product of p/(p - 2) over the odd primes

I'd like to calculate, or find a reasonable estimate for, the Mertens-like product $$\prod_{2<p\le n}\frac{p}{p-2}=\left(\prod_{2<p\le n}1-\frac{2}{p}\right)^{-1}$$ Also, how does this behave ...
1
vote
1answer
239 views

Proving $\pi(\sqrt{p_{1}p_{2}\cdots p_{n}})>2n$ for $n \geq 6$

I am having trouble in solving the following problem. Let $p_{n}$ denote the $n$-th prime. Then prove that $$\pi(\sqrt{p_{1}p_{2}\cdots p_{n}})>2n$$ for $n \geq 6$. No idea how to start.
34
votes
6answers
2k views

How hard is the proof of $\pi$ or e being transcendental?

I understand that $\pi$ and e are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
6
votes
1answer
760 views

Why is the following evaluation of Apery's Constant wrong and do you have suggestions on how, if at all, this method could be improved?

Please let me summarize the method by which L. Euler solved the Basel Problem and how he found the exact value of $\zeta(2n)$ up to $n=13$. Euler used the infinite product $$ \displaystyle f(x) = ...
10
votes
5answers
580 views

Bounding the integral $\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}}$

If $x \geq 2$, then how do we prove that $$\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}} = O\Bigl(\frac{x}{\log^{n}{x}}\Bigr)?$$