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

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1
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1answer
54 views

Additive properties of sequences: trying to understand Schnirelmann density

I have started reading Gelford & Linnik's elementary methods in analytic number theory (1965). They define a sequence $A$ of integers as: $$0, a_1, a_2,a_3,\dots$$ where $$0 < a_1 < a_2 ...
3
votes
2answers
126 views

Proving all sufficiently large integers can be written in the form $a^2+pq$.

This is one of those numerous questions I ask myself, and to which I seem unable to answer: Can every integer greater then $657$ be written in the form $a^2+pq$, with $a\in\mathbb Z$ and $p,q$ ...
2
votes
1answer
96 views

Euler Totient Issues

I was skimming again through Dummit & Foote's Abstract Algebra and I came across this exercise: Prove that for any given positive integer $N$ there exist only finitely many integers $n$ with ...
4
votes
2answers
139 views

Asymptotic behavior of $\sum_{n>x} \frac{\log n}{n^2}$

There is a well-known question that seeks the asymptotic behaviour of this function, for $x\geq 2$: $$\sum_{n\leq x} \frac{\phi(n)}{n^2}.$$ See, for example, Apostol "Introduction to Analytic Number ...
4
votes
1answer
82 views

What motivated Rademacher's contour along the Ford circles when he used the circle method?

After Ramanujan and Hardy found the infinite sum representation of the partition function $p(n)$, Rademacher went about simplifying their proof; the form generally seen involves integrating ...
2
votes
0answers
55 views

Can anyone sketch an outline of Iwaniec's proof for the upper bound regarding the Jacobsthal function?

A proof by H. Iwaniec in 'On the problem of Jacobsthal, Demonstratio Math. 11, 225–231, (1978)' shows that: $$j(N) \ll \log \log (N)$$ where $j(N)$ is the Jacobsthal function. I am very interested ...
-1
votes
1answer
94 views
3
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3answers
119 views

What does the integer span of one irrational, and one (possibly irrational) real number look like in $\mathbb{R}$?

My title was rejected a few times, here is what it was initially: If you take two real numbers- one irrational and one possibly irrational - how close does their $\mathbb{Z}$ span come to any ...
1
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0answers
44 views

Trouble computing a sum of Dirichlet characters.

Let $\chi(n)$ be a character mod $m$, and let $\rho$ be an $h$th root of unity. I am trying to compute the following sum \begin{equation} \sum_{\chi}(\rho^{-1}\chi(a) + \rho^{-2}\chi(a^2) + \cdots + ...
4
votes
1answer
91 views

Bernoulli number type asymptotics

I find an interesting formula but I can not prove it. Show that $$I_n=(-1)^{n+1}\int_0^1 B_{2n+1}(x)\cot(\pi x) \, dx\sim\frac{2(2n+1)!}{(2\pi)^{2n+1}}$$ where $B_n(x)$ is the Bernoulli Polynomials.
30
votes
3answers
617 views

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

Prove that $$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$ ($H_n=\sum_{k=1}^{n}\frac{1}{k}$)
1
vote
1answer
59 views

Lower bound for second Chebyshev function

I was wondering is there any simple way to find nice lower bound for second Chebyshev function given by formula: $$\psi(x)=\sum_{p\le x} \left\lfloor \frac{\ln x}{\ln p} \right\rfloor\ln p$$ that is ...
14
votes
3answers
644 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? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
5
votes
2answers
75 views

GCD and the Riemann zeta funtion

I'm completely stuck on this one, as I'm just starting with analytic number theory: How to write $$\sum_{a\in\mathbb{N}}\sum_{b\in\mathbb{N}}\frac{(a,b)}{a^sb^t}$$ in terms of the Riemann zeta ...
15
votes
3answers
428 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. :|
3
votes
1answer
127 views

Zero's of alternating zeta function

Is there an intuitive explanation for why all the known non-trivial solutions of $$\frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots=0$$ have a real part of $\frac{1}{2}$?
21
votes
1answer
317 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 ...
1
vote
1answer
69 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
26 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
52 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
67 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 ...
5
votes
0answers
138 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
93 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 ...
2
votes
1answer
93 views

Value of $\sum_{n=2}^{\infty}\frac{(-1)^n}{n^k}\zeta(n)$

I know that: $$\displaystyle \sum_{n=2}^{\infty}\frac{(-1)^n}{n}\zeta(n)=\gamma$$ Is there any known value for $\displaystyle \sum_{n=2}^{\infty}\frac{(-1)^n}{n^k}\zeta(n)$ for any ...
8
votes
1answer
192 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 ...
6
votes
0answers
107 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
75 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 ...
5
votes
2answers
97 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 ...
0
votes
1answer
190 views

Regarding Chebyshev's theta function

It is known that $x\sim \theta \left ( x \right )$, where $$\theta \left ( x \right )= \sum_{p\leqslant x}\log p.$$ For all values of x for which it has been calculated, $x> \theta \left ( x ...
2
votes
1answer
98 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
92 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 ...
12
votes
0answers
221 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
47 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
36 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
123 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 $$ ...
1
vote
1answer
51 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
122 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
100 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
66 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
137 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 ...
0
votes
1answer
52 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
136 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
103 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
70 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
177 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 ...
1
vote
1answer
104 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
103 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}= ...
2
votes
2answers
158 views

Arithmetical functions summation

Problem (7.4.15) of Burton's Elementary Number Theory has been request that prove the following equalities. In this book isn't expressed Dirichlet multiplication and Riemann's zeta function before ...
4
votes
1answer
79 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) ...
5
votes
1answer
154 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 ...