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

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2
votes
0answers
184 views

Entropy Rate of a sequence of Random Variables with Distributions related to Primes

Let us consider a stochastic process $\mathcal{X}=\{X_i\}_{i \in \mathbb{N} }$ where $X_i$'s are independent and $X_i$ is distributed as $$X_i=p_k \ \mbox{w. p.}\frac{p_k}{\sum_{l=1}^{i}p_l},\ 1\leq ...
1
vote
1answer
35 views

Proving that $\sum_{i=2}^{M}\frac{\pi(x^{1/i})}{i}=O(x^{1/2})+O(Mx^{1/3})$

How do I prove that $$\sum_{i=2}^{M}\frac{\pi(x^{1/i})}{i}=O(x^{1/2})+O(Mx^{1/3}).$$ I tried to use Prime Number theorem for $\pi(x)$ and then approximating the summation by integral, but when I used ...
4
votes
1answer
93 views

Primes in binary

Let $$S_n(k)=\{1\leq m\leq n: m\ \mbox{has $k$ ones in its binary representation and $m$ is prime}\}\ \\ \forall \ n\geq 2^k-1,\ k\geq1.$$ Let $\pi(x)$ be the prime number function. Then what can be ...
19
votes
1answer
446 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: ...
11
votes
2answers
758 views

A problem about the largest prime factor of $n^2+1$

Let $f(n)$ be the largest prime factor of $n$. The image of function $g(n)=\sqrt{f(n^2+1)}$ is like this: Question: If we want to draw a horizontal line which bisects the points from $n=1$ to ...
0
votes
0answers
98 views

estimate $\sum_{x<p\le x+y} \log{p}/p$

In his paper the prime number theorem via the large sieve, A. Hildebrand made use of the following inequality $$\sum_{x<p\le x+y} \frac{\log{p}}{p} \le (2+o(1))\log{\frac{x+y}{x}}$$ where $x\ge y$ ...
2
votes
1answer
75 views

Question on Wolstenholme's theorem

In one of T. Apostol's student textbooks on analytic number theory (i.e., Introduction to Analytic Number Theory, T. Apostol, Springer, 1976) Wolstenholme's theorem is stated (Apostol, Chapt. 5, page ...
2
votes
0answers
106 views

Duffin-Schaeffer theorem/conjecture (counter)example

By the "easy" direction of Duffin-Schaeffer conjecture, it is known that if (*)$\sum_{q=1}^{\infty}\phi(q)f(q) < \infty $ (when $\phi(q)$ is euler totient function) then almost all numbers are not ...
3
votes
0answers
154 views

question about riemann zeta function

How can one prove that $$\zeta (2n)=\frac{(-1)^{n-1}2^{2n-1}\pi ^{2n}B_{2n}}{(2n)!}$$ where $n\in N$ and how can one prove that $$\zeta (2n)=\frac{(-1)^{n}2^{2n-2}\pi ...
1
vote
0answers
68 views

Order of summation of Moebius function with summations of fractional parts as coefficients

I want to show that $\displaystyle\sum_{i=0}^n\left(\mu(i)\sum_{j=1}^{\lfloor\frac{n}{i}\rfloor}\{jx\}\right)=O(n)$ for $x\in (0,1)$. I have tried to use the result that ...
3
votes
2answers
132 views

Order of a function related to divisors

Let $f(n)=\max(\{d(ab):\ a,b\le n\})$ where $d(m)$ is the number of divisors of $m.$ What is the order of $f$? In particular I'm looking for an asymptotic upper bound.
5
votes
0answers
380 views

Maximum length of sequence of non-coprimes of $N$ - least upper bound for Jacobsthal's function

I am looking at the length of the longest sequences of adjacent integers that are not coprime to $N$ for very large $N$. Let $F_N$ be the set of integers less than $N$ which are not coprime with $N$: ...
2
votes
1answer
217 views

Challenging the Chebychev function / prime number theorem?

The prime number theorem accords with the following equation for the first Chebychev function that: $$\lim_{x\rightarrow\infty}\frac{\vartheta(x)}{x}=1 \qquad (1)$$ According to Muñoz García, E. and ...
5
votes
1answer
126 views

Number of prime factors of difference of two numbers

As is the custom, define $\omega(m)$ to be number of distinct primes dividing $m$. Also, let $P(m)$ represent set of primes divisors of $m$. Let $S=\{p_1,p_2,\ldots,p_n\}$ be a set of $n$ distinct ...
5
votes
0answers
131 views

Inverting the Riemann zeta function in $s>1$

Let $s>1$ be a positive real and the Riemann zeta fucntion be defined for $s>1$ as $$ \zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}. $$ I am looking for an inversion formula for the zeta ...
0
votes
1answer
66 views

How to introduce an integer function into $\zeta$ function instead of $n$

I have a problem that I am struggling with since long and probably it is simple but I can not get through. So your help would be very welcome. Known that Riemann $\zeta$ function is defined as sum ...
2
votes
1answer
140 views

The relation of $\zeta$-function and $p^k$ for $Re(s) \le 1$?

The von Mangoldt function: $$\Lambda(n) = \begin{cases} \log p &; \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1, \\ 0 &; \mbox{otherwise.} \end{cases}$$ establishes a ...
3
votes
1answer
183 views

Analytic continuation of Riemann Zeta funtion

I am reading about zeta function from book by Ingham. In that book the following theorem is given. I am unable to understand what does he mean by finite part of plane in the statement.
5
votes
0answers
148 views

Best upper bound on the number of divisors of $n$ that are larger than $N$.

I am looking for the best upper bound on $$\sum_{\substack{d | n\\ d \geq N}} 1.$$ I know that $$ d(n) = \sum_{\substack{d | n}} 1 \leq e^{O(\frac{\log n}{\log \log n})}. $$ For my application, I ...
2
votes
1answer
68 views

Lemma from arithmetic functions

Let $f$ arithmetic and $$H(f)=\lim_{x\rightarrow \infty}\frac{1}{x\log x}\sum_{n\leq x}f(n)\log n,$$ Then $H(f)$ exists if and only if $M(f)$ exists, and $M(f)=H(f)$ Where $$M(f)=\lim_{x\rightarrow ...
10
votes
3answers
302 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 ...
7
votes
1answer
82 views

Prove that the non-trivial root of $\sum_{k=1}^{2n} p_kx^k=0$ tends to $-1$

I looked at $$ \sum_{k=1}^{2n} p_kx^k=0, $$ where $p_k$ is the $k$th prime. I found that, next to the trivial root $x_0=0$, there is only one more root $x_n$ that tends towards $-1$, when $n$ ...
0
votes
2answers
75 views

Why is $\frac{1}{2\pi i} \int_C \left( \frac{x}{n} \right)^s \frac{ds}{s} = \theta(x-n) $?

I'm trying to understand the equation: $$\frac{1}{2\pi i} \int_C \left( \frac{x}{n} \right)^s \frac{ds}{s} = \theta(x-n).$$ Here $x\in \mathbb{R}, x\geq 0$, and $C = \{s:\operatorname{Re}(s) = ...
3
votes
1answer
220 views

Proving equivalences between prime counting functions.

If we have that: $$\theta(x)=\sum_{p\leq x}\log p,$$ and $$\psi(x)=\sum_{n\leq x}\Lambda(n)$$ Where $\Lambda(n)=\log p $ if $n=p^m$ and $\Lambda(n)=0$ in another case. How can I prove that : 1) ...
10
votes
1answer
318 views

some standard estimates in Yitang Zhang's paper

I'm trying to understand Zhang's paper on prime gaps, but I can't figure out some "standard" estimates for which Zhang omitted details. As a layman in analytic number theory, I really need some hints ...
2
votes
1answer
265 views

Functional equation for Hecke $L$-series

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, Theorem II.10.3, we have Let $L(s,\psi)$ be the Hecke $L$-series attached to the Größencharakter $\psi$. Then $L(s,\psi)$ has ...
1
vote
1answer
163 views

definition of divisor functions

I have a question about the definition of divisor functions when I was reading primes in tuples by Goldston, Pintz, and Yıldırım: Let $\omega(q)$ denote the number of prime factors of a squarefree ...
8
votes
2answers
551 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) = ...
1
vote
2answers
90 views

Routine question about derivatives of automorphic forms being L^2

I consider Automorphic forms on $G = SL_2 (\mathbb{R})$, which are $\Gamma$-invariant, $K$-finite, $Z(g)$ finite, and of moderate growth. If I have such an automorphic form, which happens to be in ...
8
votes
1answer
856 views

Understanding Zhang's result of bounded prime gaps

Here is a link on the internet: https://www.dropbox.com/s/su3uak2a057yrqv/YitangZhang.pdf Can someone teach me how to use trivial estimation to reach (6.1) on page 24? Namely, how to impose ...
3
votes
1answer
132 views

Proving that Bombieri's Theorem implies Linnik's theorem

I'm stuck on a line in the proof of Bombieri implies Linnik, where Bombieri: For primitive $\chi$ mod $q$ with $q \leq T$ we define $$N(\alpha, T; \chi)=\#\{\rho=\beta+i\gamma \;:\; ...
0
votes
1answer
153 views

Vanishing of Dirichlet Series

Suppose the function $\sum_{n=1}^{\infty}{a_{n}n^{-s}}$ is $0$ on some open set $U\subset\mathbb{C}$. (Can assume the sum converges absolutely on $U$.) Is it true that $a_{n}=0$ for all $n$? (This ...
4
votes
1answer
134 views

How does it follow $s\int_1^{\infty}\frac{\psi(x)}{x^{s+1}}dx$?

I have two relations: 1)$-\frac{\zeta'(s)}{\zeta(s)}=\sum_{1}^{\infty}\frac{\Lambda(n)}{n^s}$. 2)$\psi(x)=\sum_{n\leq x}\Lambda(n)$. From these two how does it follow that ...
1
vote
1answer
260 views

Proving the Bernoulli number relation $(1+B)^n=B^n$

We know that we can generate the Bernoulli numbers using the relation $(1+B)^n=B^{[n]}$ where $B_n$ is $n$th Bernoulli number. But how we can prove this works? Thanks to all. Edit 2: is there a ...
9
votes
1answer
441 views

Sum of square root of primes

I was playing around with prime numbers and a question came into my mind: Let $S(n)$ denote the sum of square roots of primes from $2$ to the $n$th prime number. Are there infinitely many numbers $n$ ...
4
votes
1answer
107 views

Trying to understand Theorem 2.27 in a recent paper on the Chebyshev function

In February 2013, Sadegh Nazardonyavi and Semyon Yakubovich posted on arxiv: Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$. I have a question about Theorem 2.27 on page 22. My ...
7
votes
1answer
229 views

Sum of rational numbers given some properties

Let $R(n)$ denote the sum of all positive rational numbers whose numerators and denominators are less than or equal to $n$ and have no common factors. I have estimated this sum to be $$ \begin{align*} ...
2
votes
2answers
99 views

Analytic method for number theory-do we have to assert second-order logic?

I am an undergraduate. I am just starting to study logic and analytic number theory at the same time, so please forgive me if I made an elementary misunderstanding. A lot of theorem in number theory ...
4
votes
1answer
376 views

Binary vs. Ternary Goldbach Conjecture

Is there an "understandable" explanation of why the ternary Goldbach conjecture is tractable with current methods, while the binary Goldbach conjecture seems to be out of scope with current ...
5
votes
2answers
477 views

effective version of Mertens Theorem for the Euler product

I'm referring to the theorem given here, which is $$\displaystyle\lim_{n\to \infty} \:\: \left(\frac1{\ln(n)} \cdot \left(\displaystyle\prod_{p\leq n} \frac1{1-\frac1p}\right)\right) \;\;\; = \;\;\; ...
1
vote
0answers
63 views

Prime-Like sets

I need some examples of "prime-like" sets of numbers. May be this term is already known by some other standard name. Let me define it. A set $S=\{s_1,s_2,\ldots ,s_n\}\subset \mathbb{R}$, is called ...
3
votes
1answer
104 views

Can the Möbius inversion formula be applied to the second Chebyshev function?

Is this a valid application of the Möbius Inversion Formula: Define: $$\psi\left(x\right) = \sum\limits_{p^k \le x} \log p$$ So that: $$\log x! = ...
0
votes
4answers
162 views

Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1} \frac{1}{p_n}$ convergent? [duplicate]

Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1}\frac{1}{p_n}$ convergent?
4
votes
1answer
518 views

Perron's formula (Passing a limit under the integral)

I want to understand why assuming that $\sum_{n \ge 1} \frac{a_n}{n^s}$ converges uniformly for $\mathrm{Re}(s) > \sigma > 0$ with $c > \sigma$ implies that $$ \sum_{n \le x} \, \!\!^* a_n = ...
1
vote
0answers
70 views

Consequencesof the Hadamard product expression of $L(s, \chi)$

I'm going through my lecture notes and I'm stuck on the proof of For any $t>0$ and primitive $\chi$ modulo $q$ $$\sum_{\rho=\beta+i \gamma: \Lambda(\rho, ...
6
votes
1answer
148 views

analytic number theory, troubling bound on sum of $\varphi(n)$

I'm very confused about this bound, please give me any suggestions on how to prove it. (Note: $a \ll b$ is just a neater way to write $a = O(b)$) I am starting with the bound $$f(n) \ll ...
6
votes
3answers
198 views

What is the set $\{x\in\Bbb R\mid \forall q\in\Bbb Q: q^x\in\Bbb Q\}$?

What is the set $\{x\in\Bbb R\mid \forall q\in\Bbb Q: q^x\in\Bbb Q\}$? Of course $\Bbb Z$ is a subset of this set. Are there any other? if not what is the proof? is there a good reference for it?
17
votes
5answers
1k 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.
10
votes
3answers
954 views

Size of largest prime factor

It is well known and easy to prove that the smallest prime factor of an integer $n$ is at most equal to $\sqrt n$. What can be said about the largest prime factor of $n$, denoted by $P_1(n)$? In ...
1
vote
0answers
36 views

Using Gamma function to show the limiting case of Gordon's continued fraction as q approaches i.

A question similar to: How to derive the Golden mean by using properties of Gamma function? The limiting case of Gordon's continued fraction when $q$ approaches $i$ yields: $$\sqrt2 + 1 = ...