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

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24
votes
1answer
481 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 general/...
1
vote
1answer
88 views

Topologic Entropy of the the free-square flow S

someone knows how to prove that $\eta(n)= \mu^2(n)$ and a-fortiori μ(n), is not deterministic? I'm prove that the system associated to the flow $(X_S,T_S)$ is ergodic, where $X_S$ ís a closure the $...
1
vote
0answers
34 views

Diophantic Inequality. davenport Theorem

I'm studying the topic sums over primes, but I had a problem when studying the outcome of Davenport, $$\sum_{n\leqslant x} \mu(n) e(\alpha n) = O(x(\log x)^{-A})$$ more exactly, in a diophantic ...
1
vote
1answer
104 views

Estimate of the logarithmic derivative of the Riemann zeta function

How I can achieve this result. If $\sigma > 1$ $$-\frac{\zeta'}{\zeta}(\sigma) \ll (\sigma -1 )^{-1}$$ Thanks!
2
votes
2answers
75 views

Orthogonality de Möbius

Does anyone know how prove that $$\sum_{n\leqslant x}\mu(n)\xi(n) =o(x)$$ when $\xi(n)$ is a multiplicative functions? I found one commentary that exist a connection of this problem with the Theory of ...
10
votes
1answer
235 views

Riemann zeta function and Bernoulli function

I encountered the following problem: Show that $$\zeta(2n+1)=\frac{(-1)^{n+1}(2\pi)^{2n+1}}{2(2n+1)!}\int_0^{1}B_{2n+1}(x)\cot({\pi}x)dx$$ where $B_{2n+1}(x)$ is the Bernoulli polynomial. This ...
1
vote
1answer
193 views

Asymptotic behavior of Chebyshev functions

Let $\vartheta(x)=\sum_{p\le x} \log p$, $ \psi(x) = \sum_{p^k\le x}\log p$ be chebyshev functions. I want to show that if $\vartheta(x) \sim x$, then $\psi(x) \sim x$. ($f(x)\sim g(x)$ means that $\...
8
votes
1answer
224 views

Prime Number Theorem in $\mathbb{F}_p[x]$

What is the probability that a randomly chosen monic polynomial of large degree $n$ in $\mathbb{F}_p[x]$ is irreducible? We can interpret this probability as $\displaystyle\lim_{n\to\infty}\frac{N_p(n)...
7
votes
0answers
133 views

Can we use $n\log n$ instead of $n$-th prime?

Denote $\pi(x)$ be the number of primes $\leq x,$ $p(n)$ be the $n$-th prime number. We have $\pi(p(n))=n.$ It's well known that $$\pi(x)\sim \frac{x}{\log x} \\p(n)\sim n\log n.$$ Is it always ...
1
vote
1answer
109 views

An upper bound for $-\frac{\zeta'}{\zeta}(s)-\frac{1}{s-1}$

Let $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$. We have $\frac{\zeta'}{\zeta}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}$ for $s>1$, where $\Lambda$ stands for the von Mangoldt function (http:/...
5
votes
2answers
128 views

Verifying a relation involving Bernoulli polynomials

I would appreciate help, please, as to how to verify this relation from Kato's "Fermat's Dream" p.96. He say: By the definition of $B_n(x)$, the Bernoulli polynomial, we have $$\sum_{n=0}^{\infty}\...
2
votes
1answer
118 views

Probability property that the longest side of primitive Pythagorean triples is prime

If we consider the set of the first $n$ primitive Pythagorean triples, then the probability that the triple's longest side is prime is approximately $\dfrac{1}{\log_{11.475}n}$ based on Mathematica’s $...
2
votes
1answer
230 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 ...
18
votes
0answers
457 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 ...
4
votes
0answers
63 views

How find this sum of in analytic numbers theory?

find the summion $$\sum_{p\le x}\dfrac{1}{(p-1)^2}\sum_{m=1}^{p-1}\sum_{\chi{(p)}}\sum_{a=1}^{p-1}\chi^m{(a)}e\left(\dfrac{a}{p}\right)$$ this problem is my friend gave me a question, he ...
1
vote
0answers
49 views

Compute $\phi^{-1}(k)$, $\phi$ Euler's totient function? [duplicate]

Given a positive integer $k$, I'd like to be able to compute the set of positive integers $m$ such that $m$ is prime to precisely $k$ positive integers less than $m$. In other words, I'd like to ...
2
votes
3answers
174 views

Bernoulli numbers: comparison to factorials

I am trying to understand the behaviour of the Bernoulli numbers with respect to factorials, specifically I'd like to know whether it is true that, for all $n \in N$ with $n \ge 2$ we have $$ \left|\...
1
vote
1answer
84 views

Weak Version of Dirichlet's Theorem

I was asked to prove the following: For any given $(a,b) = 1$ and $m > 0$ there are infinitely many integers $x$ such that $(a+bx,m) = 1$. Now, I have a proof worked out that involves the ...
5
votes
2answers
150 views

Looking for explanation of bound on Dirichlet's L-Function

I am reading Stein and Shakarchi's Fourier Analysis text and the proof Dirichlet's theorem and I am looking for clarification on how he derives the following for large $s$, $\lim_{s\to\infty}$ and $\...
0
votes
1answer
158 views

Is there only one analytic continuation of the Riemann zeta function?

If I were to manipulate the zeta function in a 'new way' would I end up with an analytic continuation that is equal to the one know or something completely new for values less than 1 and complex ...
6
votes
1answer
73 views

Are there asymptotically more nonabelian groups of order $p^k$ than there are abelian groups of order $\leq p^k$?

Let $\alpha(n)$ denote the number of isomorphism classes of abelian groups of order $n$ and $\alpha^\prime(n)=\sum_{m=1}^n\alpha(m)$. Similarly, define $f(p^k)$ to be the number of isomorphism ...
6
votes
1answer
211 views

Questions regarding the Riemann-Siegel $\theta$ Function

My questions are a request, please, for help in understanding some comments in the wikipedia article discussing the Riemann-Siegel $\theta$ function http://en.wikipedia.org/wiki/Riemann%E2%80%...
0
votes
1answer
62 views

Where are the resources on the prime number theorem?

I am looking for resources which explain the prime number theorem to 18 year old students. I am not seeking a proof of the result but something which will have an impact and motivate a student to ...
2
votes
1answer
304 views

Asymptotics for the divisor function

I am attempting to understand Tao's post of 23 September 2008 given here concerning the divisor bound. My troubles are when he uses the big-O notation in proving what he lists as bound (4) $$ d(n) \...
2
votes
0answers
107 views

Integer values of the Riemann function - II

For what value of $n \ge 2$ can we have an real $x > 0$ such that both the numbers $$ \zeta\Big(1+\frac{1}{x}\Big) \text{ and } \zeta\Big(1+\frac{1}{nx}\Big) $$ are positive integers.
0
votes
0answers
82 views

Count of numbers with the given prime factors in a range [duplicate]

Given two primes: $p$ and $q$, $p \neq q$ and $n \in N$ find count of numbers $u$, so that $u \leq n$ and $u = p^k q^l$; $k, l \in N$. If we'd given with just one prime $p$ this count would be ...
10
votes
2answers
301 views

Numbers divisible by the square of their largest prime factor

Let $p(n)$ be greatest prime factor of $n$, denote $A=\{n\mid p^2(n)\mid n,n\in \mathbb N\}.$ $A=\{4,8,9,16,18,25,27,32,36,49,50,\cdots\},$ see also A070003. Define $f(x)=\sum_{\substack{n\leq x\\n\...
1
vote
1answer
209 views

Estimating the upper bound of prime count in the given range

I need to estimate count of primes in the range $[n..m)$, where $n < m$, $n \in N$ and $m \in N$ and this estimation must always exceed the actual count of primes in the given range (i.e. be an ...
4
votes
3answers
174 views

Two questions regarding $\mathrm {Li}$ from “Edwards”

I would appreciate help understanding a relation in Edwards's "Riemann's Zeta Function." On page 30 he has: $$\int_{C^{+}} \frac{t^{\beta - 1}}{\log t}dt = \int_{0}^{x^{\beta}}\frac{du}{\log u}= \...
10
votes
3answers
359 views

Arithmetical Functions Sum, $\sum_{d|n}\sigma(d)\phi(\frac{n}{d})$ and $\sum_{d|n}\tau(d)\phi(\frac{n}{d})$

$$\sum_{d|n}\sigma(d)\phi\left(\frac{n}{d}\right)=n\tau(n) ,\\ \sum_{d|n}\tau(d)\phi\left(\frac{n}{d}\right)=\sigma(n)$$ The problem (7.4.15) of Burton's Elementary Number Theory has been request to ...
4
votes
1answer
224 views

Partial summation: integral version

In a book about analytic number theory, I found two lemmas about partial summation. The first one is the discrete version of partial summation (See http://en.wikipedia.org/wiki/Summation_by_parts) ...
7
votes
1answer
218 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 ...
3
votes
1answer
164 views

Is there an explicit formula connected to $(\log\zeta(s))^2$?

Riemann used $\log\zeta(s)$ and, essentially, Perron's formula to find the explicit formula for his prime counting function, $\Pi(n)$: $li(x)-\displaystyle\sum_{\rho}li(x^\rho)-\log 2-\int_{x}^{\...
1
vote
2answers
855 views

Abscissa of Convergence (and of Absolute Convergence) of the Derivative of a Dirichlet Series

Given the series: $$F(s) = \sum f(n) n^{-s}$$ with abscissa of convergence $\sigma_c$. It's derivative would be: $$F'(s) = - \sum_{n = 1}^\infty \frac{f(n) \log(n)}{n^s}$$ Aopstol, "Intro to ...
8
votes
4answers
648 views

The asymptotic expansion for the weighted sum of divisors $\sum_{n\leq x} \frac{d(n)}{n}$

I am trying to solve a problem about the divisor function. Let us call $d(n)$ the classical divisor function, i.e. $d(n)=\sum_{d|n}$ is the number of divisors of the integer $n$. It is well known that ...
6
votes
1answer
443 views

A general explicit formula for the generalized divisor summatory function?

Mertens function has, by residues, an explicit formula of $M(x)=\displaystyle\sum_{\rho}\frac{x^\rho}{\rho\zeta'(\rho)}-2+\sum_{n=1}^\infty\frac{(-1)^{2 n}(2\pi)^{2n}}{(2n)! n \zeta(2n+1)x^{2n}}$ ...
7
votes
0answers
115 views

Weierstrass product expression for Klein's j-invariant

The first sentence of @ccorn's answer to a previous question of mine was: “Because of the modular symmetries of $j(\tau)$, the zeros of $j(\tau)$ are precisely the $\operatorname{SL}(2,\mathbb{...
4
votes
2answers
525 views

Some identities with the Riemann zeta function

Can someone either help derive or give a reference to the identities in Appendix B, page 27 of this, http://arxiv.org/pdf/1111.6290v2.pdf Here is a reproduction of Appendix B from Klebanov, Pufu, ...
2
votes
1answer
96 views

Between Mertens' theorems

It is well-known that $$ \sum_{p\le x}\frac{\log p}{p}=\log x+O(1) $$ and $$ \sum_{p\le x}\frac1p=\log\log x+M+o(1). $$ What is the order of $$ \sum_{p\le x}\frac{\sqrt{\log p}}{p} $$ ?
3
votes
0answers
795 views

Proofs of trivial zeros of zeta function?

I know that the trivial zeros of zeta function are negative even integers . I have seen the wiki-proof using the functional equation of zeta function, I might have seen a proof using Bernoulli ...
7
votes
1answer
509 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 ...
5
votes
1answer
326 views

Applying Möbius Inversion to $\Pi(x)$ and $\pi(x)$

I would appreciate help as to how to apply the Möbius inversion theorem to prime counting $\Pi (x)$ and $\pi (x)$, where: $$\Pi(x) := \sum_{n = 1}^\infty \frac{1}{n} \pi(x^{1/n})$$ and $\pi (x) = \...
3
votes
1answer
195 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 ...
5
votes
1answer
120 views

Help using the inclusion-exclusion principle to prove $\sum_{n|d}\mu(d)\frac{n}{d}=\varphi(n)$

$(1)$Using the Dirichlet convolution theorem, $$\sum_{n|d}\mu(d)\frac{n}{d}=\sum_{ab=n}\mu(a)b$$ Let $n=p_1\cdot p_2\cdots p_n\cdot k$, where $k$'s prime factorisation contains $p_k$ only if $ k\in\{...
6
votes
2answers
98 views

Prove that $\sum_{n=1}^\infty \frac{\sigma_a(n)}{n^s}=\zeta(s)\zeta(s-a)$

I would appreciate a hint concerning how to surpass the roadblock I've encountered in my attempt at a proof below. A nicer proof than mine would also help (Edit: The latter part is now done by Gerry ...
2
votes
0answers
160 views

Elementary Proof of Landau's count on number representable as sum of two squares

In Analytic Number Theory by Iwaniec and Kowalski, there is an elementary proof of Landau's result of $\#\{n \le x: \exists a, b,\ s.t.\ n = a^2 + b^2\} \sim Cx/\sqrt{\log x}$ with an explicit ...
3
votes
0answers
81 views

Goldbach weak conjecture verification

I found from http://en.wikipedia.org/wiki/Talk:Goldbach%27s_weak_conjecture that Goldbach's weak conjecture might have been proven but the proof has not been peer reviewed yet. What results of the ...
1
vote
1answer
105 views

Prove that the number of primes satisfying this is $\log(n)$

Let $\textrm{ord}(n)$ denote the number of primes $p$ such that order of $10$ modulo $p$ is $n$. Prove that $\textrm{ord}(n) \sim \log(n)$.
2
votes
0answers
45 views

Apostol ANT chapter 13 Question 9

Given $L(s,\chi)$ has a zero of order $m\ge1$ at $s=1+it$, prove that for this t we have: (a) $\frac{L'}{L}(\sigma+it,\chi)=\frac{m}{\sigma-1}+O(1)$ as $\sigma \to 1^{+}$ and (b) there exists an ...
1
vote
2answers
102 views

Dirichlet series for prime sequence

With $p_n = n^{th}$ prime and $f(s):=\sum_{n=1}^\infty 1/p_n^s$ when the series converges. What is the status of the following questions: What is the abscissa of convergence of $f$? What are the ...