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

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34
votes
6answers
3k 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 ...
34
votes
3answers
685 views

How to prove that $\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$

Prove that $$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$ $H_n$ denotes the harmonic numbers.
34
votes
2answers
1k views

Small primes attract large primes

$$ \begin{align} 1100 & = 2\times2\times5\times5\times11 \\ 1101 & =3\times 367 \\ 1102 & =2\times19\times29 \\ 1103 & =1103 \\ 1104 & = 2\times2\times2\times2\times ...
29
votes
5answers
2k views

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

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$ The ...
28
votes
1answer
1k views

Are these zeros equal to the imaginary parts of the Riemann zeta zeros?

Edit 8.8.2013: See this question also. The Fourier cosine transform of an exponential sawtooth wave times $e^{-x/2}$: $$\operatorname{FourierCosineTransform}(\operatorname{SawtoothWave}(e^x)\cdot ...
27
votes
2answers
968 views

Are there infinitely many $x$ for which $\pi(x) \mid x$?

Let $\pi(x)$ denote the Prime Counting Function. One observes that, $\pi(6) \mid 6$, $\pi(8) \mid 8$. Does $\pi(x) \mid x$ for only finitely many $x$, or is this fact true for infinitely many ...
27
votes
3answers
535 views

Sequence of numbers with prime factorization $pq^2$

I've been considering the sequence of natural numbers with prime factorization $pq^2$, $p\neq q$; it begins 12, 18, 20, 28, 44, 45, ... and is A054753 in OEIS. I have two questions: What is the ...
27
votes
2answers
598 views

Rounding is asymptotically useless?

Recently I came across the nice result that $$\left\lfloor n \right\rfloor - \left\lfloor\frac{n}{2}\right\rfloor + \left\lfloor\frac{n}{3}\right\rfloor - \left\lfloor \frac{n}{4}\right\rfloor + ...
26
votes
3answers
1k 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 ...
26
votes
1answer
861 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 ...
22
votes
1answer
2k 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. Here is my list: The Riemann Hypothesis: A Resource ...
21
votes
4answers
624 views

Evaluating $\sum\limits_{n=1}^{\infty} \frac{1}{n\operatorname{ GPF}(n)}$, where $\operatorname{ GPF}(n)$ is the greatest prime factor

$\operatorname{ GPF}(n)=$Greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. How to evaluate convergence/divergence/value of the sum $$\sum_{n=1}^{\infty} ...
21
votes
3answers
408 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 ...
21
votes
1answer
323 views

Intuition for Class Numbers

So I've been thinking about the analytic class number formula lately, and class numbers in general and I'm trying to develop a good intuition for them. My basic question, which may be too ...
20
votes
4answers
857 views

Evaluation of $\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)$?

I would like to evaluate the sum $$ \sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right) $$ Where $\operatorname{Si}$ is the sine integral, defined as: $$\operatorname{Si}(x) := ...
20
votes
1answer
509 views

Upper bound on differences of consecutive zeta zeros

The average gap $\delta_n=|\gamma_{n+1}-\gamma_n|$ between consecutive zeros $(\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i)$ of Riemann's zeta function is $\frac{2\pi}{\log\gamma_n}.$ There are many ...
19
votes
6answers
892 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? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
19
votes
4answers
2k 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 ...
19
votes
3answers
891 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 ...
19
votes
1answer
431 views

On existence of an integer between $\sqrt{n}$ and $\sqrt{2n}$ coprime to $n$

I have one proof of the following statement, and I would like to know if there is a simpler proof. I am not sure if “simpler” is the right word or not but, for the purpose of this question, I prefer ...
16
votes
5answers
985 views

Intervals that are free of primes

How can I prove that exists intervals as large as I want that are free of primes? I mean, $\forall \ k \in \mathbb{N}, \exists \ k$ consecutive positive integers none of which is a prime.
16
votes
1answer
244 views

Are there infinite many $n\in\mathbb N$ such that $\pi(n)=\sum_{p\leq\sqrt n}p$?

Are there infinite many $n\in\mathbb N$ such that $$\pi(n)=\sum_{p\leq\sqrt n}p,\tag{1}$$ where $\pi(n)$ is the Prime-counting_function? For example, ...
15
votes
3answers
444 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. :|
15
votes
2answers
768 views

A Conjecture of Schinzel and Sierpinski

Melvyn Nathanson, in his book Methods in Number Theory (Chapter 8: Prime Number's) states the following: A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can ...
15
votes
1answer
347 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} = ...
14
votes
6answers
2k 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 ...
14
votes
2answers
965 views

Logarithmic derivative of Riemann Zeta function

Given the logarithmic derivative of the zeta function $\dfrac{\zeta^\prime (s)}{\zeta(s)}$ how does it behave near $s=1$? I mean if for $s=1$ the Laurent series for the logarithmic derivative becomes ...
14
votes
2answers
459 views

Primes sum ratio

Let $$G(n)=\begin{cases}1 &\text{if }n \text{ is a prime }\equiv 3\bmod17\\0&\text{otherwise}\end{cases}$$ And let $$P(n)=\begin{cases}1 &\text{if }n \text{ is a prime ...
14
votes
1answer
272 views

What is a zeta function?

In my readings, I've come across a wide variety of objects called zeta functions. For example, the Ihara zeta function, Igusa local zeta function, Hasse-Weil zeta function, etc. My question is simple: ...
13
votes
4answers
361 views

Asymptotic formula for $\sum_{n \le x} \frac{\varphi(n)}{n^2}$

Here is yet another problem I can't seem to do by myself... I am supposed to prove that $$\sum_{n \le x} \frac{\varphi(n)}{n^2}=\frac{\log x}{\zeta(2)}+\frac{\gamma}{\zeta(2)}-A+O \left(\frac{\log ...
13
votes
3answers
481 views

Rate of divergence for the series $\sum |\sin(n\theta) / n|$

In the following we consider the series $$ S(N;\theta)= \sum_{n = 1}^{N} \left| \frac{\sin n\theta}{n} \right| $$ parametrized by $\theta$. It is well known that this series (taking the limit ...
13
votes
3answers
2k views

Non-increasing sequence of positive real numbers with prime index

If $a_n$ is a sequence of non-increasing positive numbers, then suppose we already know that $$\sum_p a_p$$ converges, when $p$ runs over the primes, what should be used to prove that $$\sum_n ...
12
votes
2answers
257 views

An identity about the Pi and Riemann's zeta function

How to prove the following identity? $$\sum_{n=1}^{\infty}\frac{(m-1)^n-1}{m^n}\zeta(n+1)=\pi\cot\left(\frac{\pi}{m}\right)$$
12
votes
2answers
656 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. ...
12
votes
1answer
274 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. ...
12
votes
2answers
400 views

On prime factors of $n^2+1$

It is a well-known conjecture that there are infinitely many primes of the form $n^2+1$. However, there are weaker results that one can prove. For example, There are infinitely many positive ...
12
votes
1answer
464 views

What is the probability that some number of the form $10223\cdot 2^n+1$ is prime?

I (David Speyer) took the liberty of doing a fairly major rewrite of this question. I hope I haven't gone too far, but I think there is an interesting question hiding here. Sierpinski proved that ...
12
votes
1answer
526 views

Going from $\Lambda$ to a prime count

A 1997 paper of Étienne Fouvry and Henryk Iwaniec, Gaussian primes, concerns the prevalence of primes that are of the form $n^2+p^2$ for prime $p$. The asymptotic result is $$\sum_{n^2+p^2\le ...
12
votes
0answers
239 views

Using the Brun Sieve to show very weak approximation to twin prime conjecture

I recently stumbled across MIT OCW for analytic number theory. As a budding number theorist, my ears perked up and I looked through some of the material haphazardly. I don't really know much about ...
11
votes
3answers
1k 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 ...
11
votes
3answers
588 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 ...
11
votes
2answers
221 views

An upper bound for $\log \operatorname{rad}(n!)$

Let $n>1$ be an integer and let $\operatorname{rad}(n!)$ denote the radical of $n$-factorial. (The radical of an integer $m$ being, loosely speaking, the product of the prime divisors of $m$.) Can ...
11
votes
2answers
536 views

Supplemental number theory text to Montgomery and Vaughan

We already have a large list of the Best ever book on Number Theory, but I'm looking for a more targeted response for analytic number theory. Specifically, I'm taking a trip on which I may or may ...
11
votes
1answer
128 views

The probability that $\dfrac{p-1}2$ is square-free

Let $Q(x)$ denote the number of square-free integers between $1$ and $x$, we obtain the approximation $$\eqalign{ &Q(x)\approx x\prod_{p\,{\rm prime}}\left(1-\dfrac1{p^2}\right)=x\prod_{p\,{\rm ...
11
votes
0answers
324 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = ...
10
votes
5answers
595 views

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

If $x \geq 2$, then how do we prove that $$\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}} = O\Bigl(\frac{x}{\log^{n}{x}}\Bigr)?$$
10
votes
3answers
436 views

Motivation for Hecke characters

The context is the definition of Hecke Größencharakter: http://en.wikipedia.org/wiki/Hecke_character This is supposed to generalize the Dirichlet $L$-series for number fields. Dirichlet characters ...
10
votes
3answers
340 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 ...
10
votes
3answers
1k 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 $\mathbb{Q}$. ...
10
votes
3answers
227 views

Where is the fallacy in the argument using Prime Number Theorem

I am reading about Prime Number Theorem from book by Ingham. As as application of PNT I found the following theorem: Now my doubt is at the step $\frac{\log(y)}{\log(x)}\rightarrow 1$, we can say ...