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Questions tagged [analytic-number-theory]

Questions on the use of the methods of real/complex analysis in the study of number theory.

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39
votes
1answer
2k views

How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $?

Note the $ p < x $ in the sum stands for all primes less than $ x $. I know that for $ s=1 $, $$ \sum_{p<x} \frac{1}{p} \sim \ln \ln x , $$ and for $ \mathrm{Re}(s) > 1 $, the partial sums ...
18
votes
5answers
3k views

How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?

I mean the Laurent series at $s=1$. I want to do it by proving $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$, based on the integral formula given in Wikipedia. ...
4
votes
2answers
531 views

Is there a way to show that $\sqrt{p_{n}} < n$?

Is there a way to show that $\sqrt{p_{n}} < n$? In this article, I show that $f_{2}(x)=\frac{x}{ln(x)} - \sqrt{x}$ is ascending, for $\forall x\geq e^{2}$. As a result, $\forall n \geq 3$ $$\frac{...
22
votes
3answers
962 views

Calculate $\sum\limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$

Calculate $$\sum \limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$$ I use software to complete the series is $\frac{2}{27} \left(18+\sqrt{3} \pi \right)$ I have no idea about it. :|
20
votes
3answers
4k views

How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?

I saw reference to this result of Chebyshev's: $$\sum_{p\leq n} \frac{\log p}{p} \sim \log n \text{ as }n \to \infty,$$ and its relation to the Prime Number Theorem. I'm looking into an information-...
14
votes
3answers
660 views

Are there any Combinatoric proofs of Bertrand's postulate?

I feel like there must exist a combinatoric proof of a theorem like: There is a prime between $n$ and $2n$, or $p$ and $p^2$ or anything similar to this stronger than there is a prime between $p$ and $...
9
votes
1answer
560 views

Limit inferior of the quotient of two consecutive primes

I have recently read an article about the prime number theorem, in which Mathematicians Erdos and Selberg had claimed that proving $\lim \frac{p_n}{p_{n+1}}=1$, where $p_k$ is the $k$th prime, is a ...
15
votes
2answers
4k views

Sum of reciprocal prime numbers [duplicate]

How can the following equation be proven? $$ \forall n > 2 : \sum_{p \le n}{\frac1{p}} = C + \ln\ln n + O\left(\frac1{\ln n}\right), $$ where $p$ is a prime number. It's not homework; I just ...
7
votes
2answers
1k 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 x}\mu(n)\left[\frac{x}{n}\...
23
votes
2answers
3k views

least common multiple $\lim\sqrt[n]{[1,2,\dotsc,n]}=e$

The least common multiple of $1,2,\dotsc,n$ is $[1,2,\dotsc,n]$, then $$\lim_{n\to\infty}\sqrt[n]{[1,2,\dotsc,n]}=e$$ we can show this by prime number theorem, but I don't know how to start I had ...
21
votes
4answers
25k views

What is the analytic continuation of the Riemann Zeta Function

I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation. What is this other ...
28
votes
3answers
777 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 ...
14
votes
3answers
1k views

On the mean value of a multiplicative function: Prove that $\sum\limits_{n\leq x} \frac{n}{\phi(n)} =O(x) $

There is a second part of the problem posted in Proving $ \frac{\sigma(n)}{n} < \frac{n}{\varphi(n)} < \frac{\pi^{2}}{6} \frac{\sigma(n)}{n}$, from Apostol's book, but I can't figure it out. It ...
8
votes
2answers
656 views

Bounds for $\zeta$ function on the $1$-line

I was going over my notes from a class on analytical number theory and we use a bound for the $\zeta$ function on the $1$ line as $\vert \zeta(1+it) \vert \leq \log(\vert t \vert) + \mathcal{O}(1)$ ...
12
votes
3answers
3k views

Lower bound for $\phi(n)$: Is $n/5 < \phi (n) < n$ for all $n > 1$?

Is it true that : $\frac {n}{5} < \phi (n) < n$ for all $n > 1$ where $\phi (n)$ is Euler's totient function . Since $\phi(n)$ has maximum value when $n$ is a prime it follows that ...
65
votes
5answers
10k views

What is so interesting about the zeroes of the Riemann $\zeta$ function?

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$ ...
53
votes
6answers
10k 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 ...
38
votes
5answers
7k views

Riemann zeta function at odd positive integers

Starting with the famous Basel problem, Euler evaluated the Riemann zeta function for all even positive integers and the result is a compact expression involving Bernoulli numbers. However, the ...
10
votes
5answers
1k views

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

If $x \geq 2$, then how do we prove that $$\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}} = O\Bigl(\frac{x}{\log^{n}{x}}\Bigr)?$$
9
votes
2answers
4k views

What does this $\asymp$ symbol mean? (subject: analytic number theory)

I'm reading a survey article by Andrew Granville on analytic number theory. On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot below....
23
votes
3answers
2k views

On Dirichlet series and critical strips

(I'll keep this one short) Given a Dirichlet series $$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$ where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero ...
13
votes
3answers
2k views

Proving $\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$

How to prove this: $$\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$$ From Apostol's number theory text i know that $$\sum\limits_{p \leq x} \frac{1}{p} = \log{\log{...
11
votes
2answers
986 views

Importance of the zero free region of Riemann zeta function

I have heard that for improving the error term in the Prime Number Theorem, we need better and better estimates on the zero free region. I have also heard that the best possible error term comes from ...
4
votes
1answer
342 views

Asymptotic formula for $\sum_{n\leq x}\sigma(n)$ knowing $\sum_{n\leq x}\frac{\sigma(n)}{n}$

Let $\sigma(n):=\sum_{d|n}d$ be the sum of all divisors of $n$. Find the asymptotic formula for $\sum_{n\leq x}\frac{\sigma(n)}{n}$ and use it to find the one for $\sum_{n\leq x}\sigma(n)$. Here is ...
15
votes
1answer
2k views

Effective Upper Bound for the Number of Prime Divisors

Let $\omega(n) = \sum_{p \mid n} 1$. Robin proves for $n > 2$, \begin{align} \omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}}. \end{align} Is there a similar ...
13
votes
2answers
681 views

Solving an integral coming from Perron's formula

In analytic number theory, Perron's formula says that $$ \sum_{1 \leq k < n} a_k + \frac{1}{2}a_n = \int_{c - i\infty}^{c+i\infty} f(s)\frac{n^s}{s}ds, $$ where $f(s) = \sum_{k \geq 1} a_k/k^s$ ...
28
votes
3answers
2k views

Ramanujan's First Letter to Hardy and the Number of $3$-Smooth Integers

A positive integer is $B$-smooth if and only if all of its prime divisors are less than or equal to a positive real $B$. For example, the $3$-smooth integers are of the form $2^{a} 3^{b}$ with non-...
13
votes
1answer
932 views

Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function

Let $f(n)$ be a multiplicative function defined by $f(p^a)=p^{a-1}(p+1)$, where $p$ is a prime number. How could I obtain a formula for $$\sum_{n\leq x} f(n)$$ with error term $O(x\log{x})$ and ...
10
votes
1answer
2k views

Why the Riemann hypothesis doesn't imply Goldbach?

I'm interested in number theory, and everyone seems to be saying that "It's all about the Riemann hypothesis (RH)". I started to agree with this, but my question is: Why then doesn't RH imply the (...
6
votes
2answers
287 views

Understanding a very elementary property of factorials

I've seen this stated in a few places. If $$\vartheta(x) = \sum_{p\le{x}} \log (p) \qquad \psi(x) = \sum_{m=1}^{\infty}\vartheta\left(\sqrt[m]{x}\right)$$ Then $$\log(x!) = \sum_{m=1}^{\infty} \...
6
votes
3answers
797 views

Does the correctness of Riemann's Hypothesis imply a better bound on $\sum \limits_{p<x}p^{-s}$?

This is follow up question on this: How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $? There it is stated that: $$ \sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + O\left(\...
5
votes
1answer
365 views

Does the Riemann zeta function converge in higher dimensions?

It is known that the Riemann zeta function $$\displaystyle \zeta(s) = \sum_{n = 1} \frac{1}{n^s}$$ converges for all $\mathrm{Re}(s) > 1$, and admits an analytic continuation to the rest of the ...
52
votes
2answers
10k views

Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. The following are excluded: Books by mathematical ...
28
votes
1answer
900 views

What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?

We have the following evaluations: $$\begin{aligned} &\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\ &\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = \frac{1}{3}\,\...
24
votes
8answers
9k views

A good reference to begin analytic number theory

I know a little bit about basic number theory, much about algebra/analysis, I've read most of Niven & Zuckerman's "Introduction to the theory of numbers" (first 5 chapters), but nothing about ...
15
votes
3answers
4k views

How many elements in a number field of a given norm?

Let $K$ be a number field, with ring of integers $\mathcal{O}_k$. For $x\in \mathcal{O}_K$, let $f(x) = |N_{K/\mathbb{Q}}(x)|$, the (usual) absolute value of the norm of $x$ over $\mathbb{Q}$. ...
18
votes
2answers
522 views

Exploiting a Diophantine approximation of $\pi^4$ into giving a series of rationals for $\pi^4$

A note about this question: The original question asked seems likely impossible so I am really asking if we can exploit the technique below into giving us a 'nice' form for $\pi^4$. By nice form I ...
13
votes
1answer
439 views

Chebyshev: Proof $\prod \limits_{p \leq 2k}{\;} p > 2^k$

How do I prove the following: $$\prod_{p \leq 2k} \; p > 2^k \text{ with } p \in \mathbb{P}$$ I tried induction, but I didn't know how to go on because I don't have a look at all numbers. Any ...
8
votes
2answers
1k views

Generating functions and the Riemann Zeta Function

The generating function for the terms of the harmonic series: $\frac{1}{n}$ is $-\ln(1 - x)$. Does an ordinary generating function exist for the terms of the zeta function $\zeta(s) = \sum_{n=1}^\...
9
votes
2answers
393 views

Generalized PNT in limit as numbers get large

If $\pi_k(n)$ is the cardinality of numbers with k prime factors (repetitions included) less than or equal n, the generalized Prime Number Theorem (GPNT) is: $$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln ...
8
votes
2answers
1k views

Euler Product formula for Riemann zeta function proof

In class we introduced Reimann Zeta function $$ \zeta (x)=\sum_{n=1}^{+\infty} \frac{1}{n^x} $$ And we proved its domain was $D=(1,+\infty)$ Now Euler proved that $$ \zeta(x)=\prod_{p\text{ prime}...
12
votes
1answer
1k views

Rate of convergence of series of squared prime reciprocals

It is well known that $\sum_{p \text{ prime}} \frac{1}{p}$ diverges, and in fact - it behaves like log of the harmonic series: $$ \sum_{p \le x} \frac{1}{p} = \log \log x + O(1). $$ It is also well ...
6
votes
1answer
299 views

Average order of $\mathrm{rad}(n)$

Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing n. Or equivalently, $$\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ ...
4
votes
1answer
113 views

Show that $p_n^{1-\epsilon}\le n$ using PNT

Assuming PNT $$\pi(x)\sim \frac{x}{\log{x}}$$ How can we show that given any $\epsilon>0$ $$p_n^{1-\epsilon}< n,$$ for all sufficiently large $n$ ($p_n$ denotes the $n^{th}$ prime.) My work: ...
2
votes
1answer
886 views

Argument of the Riemann zeta function on Re(s)=1

I refer to the lovely answer to this question. Is there an exact formula for the argument of the Riemann zeta function? Specifically, I would like to know the arguments along the line Re$(s)=1$. Some ...
1
vote
3answers
1k views

Asymptotic formula for the logarithm of the hyperfactorial

Background: I was trying to derive an asymptotic formula for the following: $$\sum_{m\leqslant n}\sum_{k\leqslant m}(m\ \mathrm{mod}\ k),$$ which I think I succeeded in doing (I will skip some steps ...
6
votes
2answers
253 views

What is known about the 'Double log Eulers constant', $\lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$?

The Euler constant is defined as $$\gamma = \lim_{n \to \infty}{\sum_{k=1}^n\frac{1}{k}-\ln{n}}$$ Let $$q = \lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$$ I managed to prove that $$\...
1
vote
0answers
189 views

Bounding Maximal gaps with Ramanujan primes

Gaps and Maximal Gaps We define terms used in this article. A prime gap as $g_n := p_{n+1} - p_n$, and we define $g_n$ as a maximal gap, if $g_i < g_n$ for all $i < n$. Define $M_{m,g}$ to be ...
0
votes
1answer
964 views

If $(a,b)=1$ then there exist positive integers $x$ and $y$ s.t $ax+by=1$. [duplicate]

How can i prove that if $\gcd(a,b)=1$ there exist $x>0$ and $y>0$ such that $ax-by=1$?
24
votes
1answer
1k 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 ...