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

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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
132 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
141 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
184 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
69 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
306 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
221 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
270 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
165 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
565 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) = \sum_{n=1}^\...
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 $L^...
8
votes
1answer
858 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 $(d,P_0)&...
3
votes
1answer
137 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 \;:\; \Lambda(\...
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 $-\frac{\zeta'(s)}{\zeta(...
1
vote
1answer
263 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
450 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
230 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 techniques?...
5
votes
2answers
498 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
105 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! = \sum\limits_{k=1}^{\infty}\psi\left(\frac{x}{k}\...
0
votes
4answers
164 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
521 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, \chi)=0}\frac{1}{1+(t-\gamma)^2}=O(...
6
votes
1answer
152 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 \frac{n}{\log(...
6
votes
3answers
200 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
991 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 = \frac{3\...
3
votes
1answer
190 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 14\...
6
votes
1answer
158 views

Analytically continue a function with Euler product

I would like to estimate the main term of the integral $$\frac{1}{2\pi i} \int_{(c)} L(s) \frac{x^s}{s} ds$$ where $c > 0$, $\displaystyle L(s) = \prod_p \left(1 + \frac{2}{p(p^s-1)}\right)$. ...
1
vote
1answer
91 views

Why does the theta function decay exponentially as $x \rightarrow \infty$?

I'm trying to understand the proof of the functional equation for the L-series of primitive, even Dirichlet characters. For even, primitive characters we have $$\theta_\chi(x):=\sum_{n\in \mathbb{Z}} \...
5
votes
2answers
168 views

To estimate $\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$

How may we estimate $$\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$$ where for every positive integer $m$ , $d(m)$ denotes the number of positive divisors of $m$ ?
6
votes
1answer
162 views

Finding near-integers in a range

I have a (transcendental) constant $\alpha$ and a fixed parameter $\varepsilon>0.$ I'd like to find all positive integers $n<\ell$ for which $\|n\alpha\|<\varepsilon,$ where $\|x\|$ is the ...
3
votes
1answer
557 views

Definition of nebentypus in $L$-functions.

In Iwaniec and Kowalski, the term nebentypus is mentioned several times in the book. Every time it seems to just refer to a character $\chi$. Since I don't see the authors defining nebentypus, can ...
1
vote
1answer
84 views

Iwaniec Kowalski Notation

On page 532 of the book analytic number theory by Iwaniec and Kowalski, the following notation is used: $C^{~\infty}$ and $\tau(n,\chi)$. Could anyone tell me what these represent? (the former is ...
6
votes
2answers
252 views

Prime power Gauss sums are zero

Fix an odd prime $p$. Then for a positive integer $a$, I can look at the quadratic Legendre symbol Gauss sum $$ G_p(a) = \sum_{n \,\bmod\, p} \left( \frac{n}{p} \right) e^{2 \pi i a n / p}$$ where ...
1
vote
2answers
68 views

showing that $\log(N) \leq \prod_{n \leq N} {(1-p^{-1})^{-1}}$

i can't see that $H_n \leq \prod_{n \leq N}{(1-p^{-1})^{-1}}$ and i can't see that $\log(N) \leq \prod_{n \leq N} {(1-p^{-1})^{-1}}$ p is prime and $H_n$ is harmonic series
5
votes
1answer
575 views

Clarkson's Proof of the Divergence of Reciprocal of Primes

In Tom Apostol's book, he credits the proof of the divergence of the sum of reciprocal of primes to Clarkson. To begin, we assume $\{p_n\}$ is an enumeration of the primes and $$\sum_{n=1}^\infty\frac{...
7
votes
5answers
534 views

Proving $\sqrt{2}\in\mathbb{Q_7}$?

Why does Hensel's lemma imply that $\sqrt{2}\in\mathbb{Q_7}$? I understand Hensel's lemma, namely: Let $f(x)$ be a polynomial with integer coefficients, and let $m$, $k$ be positive integers ...
3
votes
0answers
146 views

Existence of zeros of Mellin transform and properties of function to be transformed

Mellin transform of function $f(x)$ defined for $x\geqslant 0$ is given by $$ f^\ast(z) =\int\limits_0^\infty x^{z} f(x) \frac{dx}{x}. $$ I consider only exponentially decreasing (there exist such ...
2
votes
1answer
148 views

Stable points and the fundamental domain of the modular group

Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq 1/2\}$ its fundamental domain. How ...