Questions on the use of the methods of real/complex analysis in the study of number theory.
22
votes
1answer
673 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 ...
16
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 ...
10
votes
5answers
467 views
Bounding the integral $\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}}$
If $x \geq 2$, then how do we prove that $$\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}} = O\Bigl(\frac{x}{\log^{n}{x}}\Bigr)?$$
6
votes
2answers
458 views
Asymptotic formula for $\mu(n)[x/n]^2$ summation
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 $$\sum_{n \le ...
3
votes
1answer
494 views
Intuition and Stumbling blocks in proving the finiteness of WC group
After reading many articles about the Tate-Shafarevich Group ,i understood that "in naive perspective the group is nothing but the measure of the failure of Hasse principle,
and coming to its ...
4
votes
1answer
103 views
Proof involving the logarithmic integral
Another exercise from Apostol's book, this time we're supposed to prove
$$\mathrm{Li}(x)=\frac{x}{\log x}+\int_2^x \frac{dt}{\log^2t}-\frac{2}{\log 2}.$$
which is easy to do via integration by ...
30
votes
5answers
2k views
How hard is the proof of $\pi$ or e being transcendental?
I understand that $\pi$ and e are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
16
votes
3answers
665 views
On Dirichlet series and critical strips
(I'll keep this one short)
Given a Dirichlet series
$$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$
where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero ...
12
votes
1answer
429 views
What is the probability that some number of the form $10223\cdot 2^n+1$ is prime?
I (David Speyer) took the liberty of doing a fairly major rewrite of this question. I hope I haven't gone too far, but I think there is an interesting question hiding here.
Sierpinski proved that ...
6
votes
3answers
647 views
Proving $\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$
How to prove this: $$\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} = ...
10
votes
2answers
309 views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of ...
3
votes
2answers
444 views
Erdős and the limiting ratio of consecutive prime numbers
The following is a piece of math lore from the late forties, which was described in an Intelligencer article entitled "The Elementary Proof of the Prime Number Theorem". It reads:
Turán, who was ...
9
votes
3answers
761 views
$\frac {n}{5} < \phi (n) < n$ 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 ...
6
votes
2answers
600 views
Sum of reciprocal prime numbers
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 ...
24
votes
1answer
844 views
Are these zeros equal to the imaginary parts of the Riemann zeta zeros?
The Fourier cosine transform of an exponential sawtooth wave times $e^{-x/2}$:
$$\operatorname{FourierCosineTransform}(\operatorname{SawtoothWave}(e^x)\cdot e^{-\frac{x}{2}})$$
can be plotted with ...
4
votes
2answers
119 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 $\vert \zeta(1+it) \vert \leq \log(\vert t \vert) + \mathcal{O}(1)$ ...
13
votes
3answers
1k 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 $$\sum_n ...
6
votes
1answer
173 views
Euler's summation by parts formula
I'm beginning analytic number theory and I see this formula in Apostol's book : If $f$ has a continuous derivative $f'$ on the interval $[y,x]$, where $0 < y < x$, then
$$
\sum_{y < n \le x} ...
23
votes
3answers
999 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 ...
7
votes
1answer
313 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 ...
7
votes
2answers
170 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:
$$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln ...
8
votes
1answer
227 views
How do I prove $\sum_{n \leq x} \frac{\mu (n)}{n} \log^2{\frac{x}{n}}=2\log{x}+O(1)$? Can I use Abel summation?
I am wondering if it is possible to solve this problem using Abel summation:
$$\sum_{n \leq x} \frac{\mu (n)}{n} \log^2{\frac{x}{n}}=2\log{x}+O(1)$$
Or maybe I am on the wrong track?
6
votes
1answer
309 views
Showing that $\log(\log(N+1)) \leq 1+\sum\limits_{p \leq N} \frac{1}{p}$
I can't see how you get this.
I want to show that
$$\log(\log(N+1)) \leq \sum_{p \leq N} \frac{1}{p}+1$$
Can't see how it follows from this. So you show that
$$0 \lt -\log(1-x)-x \lt ...
6
votes
1answer
254 views
How to derive an identity between summations of totient and Möbius functions
I have the following identities
$$\sum_{n \le x} \varphi(n) = \frac{1}{2} \sum_{n \le x} \mu(n) \left[\frac{x}{n}\right]^2 + \frac{1}{2}$$
$$\sum_{n \le x} \frac{\varphi(n)}{n} = \sum_{n \le x} ...
5
votes
2answers
375 views
An identity involving the Möbius function
$$\sum_{n=1}^{\infty}\frac{1}{n^s}\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=1$$
for $s>1$.
How do I prove this identity?
3
votes
2answers
444 views
Does the correctness of Riemann's Hypothesis imply a better bound on $\sum \limits_{p<x}p^{-s}$?
This is follow up question on this: How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \mathrm{Re}(s) < 1 $?
There it is stated that:
$$
\sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + ...
9
votes
3answers
342 views
Prove $\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 ...
6
votes
2answers
575 views
Calculating the Zeroes of the Riemann-Zeta function
Wikipedia states that
The Riemann zeta function $\zeta(s)$ is defined for all complex numbers $s \neq 1$. It has zeros at the negative even integers (i.e. at $s = −2, −4, −6, ...)$. These are ...
4
votes
1answer
186 views
Asymptotics for sums of the form $\sum \limits_{\substack{1\leq k\leq n \\ (n,k)=1}}f(k)$
How can we find an asymptotic formula for
$$\sum_{\substack{1\leq k\leq n \\ (n,k)=1}}f(k)?$$
Here $f$ is some function and $(n,k)$ is the gcd of $k$ and $n$. I am particularily interested in the case
...
4
votes
1answer
409 views
On the convergence of $\sum \mu(n)/n^s$
I arrived at something during my maths ponderings which is really exciting for me.
It is clearly stated in the book on Riemann Hypothesis by Borwein that the convergence of $\sum_{n=1}^{\infty} ...
3
votes
0answers
87 views
existence of closed forms of certain Dirichlet series
In the following post we find an interesting calculation of
$$ \sum_{n\ge 1} \frac{\mu(n)^2}{n\varphi(n)}.$$
I had been trying to do this calculation myself, setting the slightly more ambitious goal ...
3
votes
2answers
266 views
Asymptotic formula for d(n)/n summation
I was trying to show $$\sum_{n \le x} \frac{d(n)}{n} = \frac{1}{2}\log(x)^2 + 2\gamma \log(x) + O(1)$$ where $d(n)$ is the number of divisors of $n$ and $\gamma$ is the Euler constant using the ...
20
votes
5answers
814 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 ...
25
votes
3answers
371 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 ...
8
votes
2answers
386 views
Values of the Riemann Zeta function and the Ramanujan Summation - How strong is the connection?
The Ramanujan Summation of some infinite sums is consistent with a couple of sets of values of the Riemann zeta function. We have, for instance, $$\zeta(-2n)=\sum_{n=1}^{\infty} n^{2k} = 0 ...
6
votes
2answers
202 views
Question regarding Von-Mangoldt function.
Let $\psi(x) := \sum_{n\leq x} \Lambda(n)$ where $\Lambda(n)$ is the Von-Mangoldt function.
I want to show that if $$ \lim_{x \rightarrow \infty} \frac{\psi(x)}{x} =1 $$ then also $$\lim_{x\rightarrow ...
10
votes
5answers
550 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 ...
8
votes
3answers
600 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 $\mathbb{Q}$. ...
13
votes
3answers
352 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 ...
9
votes
1answer
318 views
Polar Density of a Set of Primes
In Chapter 7 of Marcus' Number Fields, he defines the polar density of a set $A$ of primes of a number field $K$ as follows:
Definition: If some $n$th power of the function
$$\zeta_{K,A}(s) = ...
8
votes
1answer
112 views
Comparing average values of an arithmetic function
Suppose $f(n)$ is a positive real-valued arithmetic function such that
$$
\frac1n\sum_{k=1}^nf(k)\sim g(n)
$$
for $g(x)$ a monotonic increasing function. What can be said about the asymptotic behavior ...
7
votes
2answers
213 views
Intuitive explanation with rigorous details why zeta has infinitely many zeros?
I have seen a proof outline that $\zeta$ has infinitely many zeros on the critical line here but what I really want is:
Simplest possible (least "magic") argument that explains why zeta has ...
6
votes
1answer
114 views
Why is width of critical strip what it is?
For Riemann zeta function and $L$-functions of number fields, the width of critical strip is $1$. For $L$-functions of modular forms of weight $k$, the width of the critical strip is $k$.
Why is ...
5
votes
0answers
106 views
Density of products of a certain set of primes
I have an infinite set S of prime numbers with relative density 0 (that is, $\lim_ns_n/p_n=\infty$ with $S=\{s_1,s_2,\ldots\}$ and $s_1 < s_2< \cdots$). I would like to find the size (in some ...
4
votes
1answer
264 views
How will this equation imply PNT
So we have $$\sum_{n \leq x} \frac{\Lambda (n)}{n}=\log{x}+C+o(1)$$ where $C$ is a constant, its partial summation is $$\sum_{n \leq x} \frac{\Lambda (n)}{n}=\frac{\psi(x)}{x}+\int_1^x \frac{\psi ...
3
votes
1answer
121 views
How to derive the Golden mean by using properties of Gamma function?
The Golden mean known as $\frac{1+\sqrt{5}}{2}$.
How could one show the Golden mean can be expressed as
$$
\frac{2\cdot 3\cdot 7\cdot 8\cdot 12\cdot 13\cdots}{1\cdot 4\cdot 6\cdot 9\cdot 11\cdot ...
0
votes
0answers
89 views
Divergence of the Derivative of the Prime Counting Function
On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written
$$
\pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1}
$$
with $ \operatorname{R}(z) = ...
20
votes
4answers
569 views
Evaluating $\sum\limits_{n=1}^{\infty} \frac{1}{n\operatorname{ GPF}(n)}$, where $\operatorname{ GPF}(n)$ is the greatest prime factor
$\operatorname{ GPF}(n)=$Greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.
How to evaluate convergence/divergence/value of the sum
$$\sum_{n=1}^{\infty} ...
8
votes
2answers
302 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 ...
6
votes
1answer
190 views
how to understand $\log\zeta(s)$ (Riemann zeta function)?
I know that if a function $f$ is analytic and has no zeros in a simple connected region, then we can define $\log{f}$ making it analytic in that region.
Let's assume $Re(s)>1$.
Is $\zeta(s)$ ...
