Questions on the use of the methods of real/complex analysis in the study of number theory.
0
votes
0answers
17 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 $$N(\alpha, T; \chi)=\#\{\rho=\beta+i\gamma \;:\; ...
0
votes
0answers
49 views
Vanishing of Dirichlet Series
Suppose the function
$\sum_{n=1}^{\infty}{a_{n}n^{-s}}$ is $0$ on some open set $U\subset\mathbb{C}$. (Can assume the sum converges absolutely on $U$.)
Is it true that $a_{n}=0$ for all $n$?
(This ...
4
votes
1answer
75 views
How does it follow $s\int_1^{\infty}\frac{\psi(x)}{x^{s+1}}dx$?
I have two relations:
1)$-\frac{\zeta'(s)}{\zeta(s)}=\sum_{1}^{\infty}\frac{\Lambda(n)}{n^s}$.
2)$\psi(x)=\sum_{n\leq x}\Lambda(n)$.
From these two how does it follow that ...
8
votes
1answer
147 views
Sum of square root of primes
I was playing around with prime numbers and a question came into my mind:
Let $S(n)$ denote the sum of square roots of primes from $2$ to the $n$th prime number.
Are there infinitely many numbers $n$ ...
4
votes
1answer
29 views
Trying to understand Theorem 2.27 in a recent paper on the Chebyshev function
In February 2013, Sadegh Nazardonyavi and Semyon Yakubovich posted on arxiv: Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$.
I have a question about Theorem 2.27 on page 22.
My ...
4
votes
0answers
69 views
Sum of rational numbers given some properties
Let $R(n)$ denote the sum of all positive rational numbers whose numerators and denominators are less than or equal to $n$ and have no common factors. I have estimated this sum to be
$$
\begin{align*}
...
2
votes
2answers
54 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
99 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 ...
3
votes
2answers
48 views
effective version of Mertens Theorem for the Euler product
I'm referring to the theorem given here, which is
$$\displaystyle\lim_{n\to \infty} \:\: \left(\frac1{\ln(n)} \cdot \left(\displaystyle\prod_{p\leq n} \frac1{1-\frac1p}\right)\right) \;\;\; = \;\;\; ...
1
vote
0answers
43 views
Prime-Like sets
I need some examples of "prime-like" sets of numbers. May be this term is already known by some other standard name. Let me define it. A set $S=\{s_1,s_2,\ldots ,s_n\}\subset \mathbb{R}$, is called ...
1
vote
1answer
27 views
Can the Möbius inversion formula be applied to the second Chebyshev function?
Is this a valid application of the Möbius Inversion Formula:
Define: $$\psi\left(x\right) = \sum\limits_{p^k \le x} \log p$$
So that: $$\log x! = ...
0
votes
4answers
74 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?
2
votes
1answer
58 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
$$
\sum_{n \le x} \, \!\!^* a_n = ...
1
vote
0answers
44 views
Consequencesof the Hadamard product expression of $L(s, \chi)$
I'm going through my lecture notes and I'm stuck on the proof of
For any $t>0$ and primitive $\chi$ modulo $q$
$$\sum_{\rho=\beta+i \gamma: \Lambda(\rho, ...
5
votes
1answer
80 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 $$f(n) \ll ...
5
votes
3answers
142 views
What is the set $\{x\in\Bbb R\mid \forall q\in\Bbb Q: q^x\in\Bbb Q\}$?
What is the set $\{x\in\Bbb R\mid \forall q\in\Bbb Q: q^x\in\Bbb Q\}$?
Of course $\Bbb Z$ is a subset of this set.
Are there any other? if not what is the proof? is there a good reference for it?
15
votes
5answers
828 views
Intervals that are free of primes
How can I prove that exists intervals as large as I want that are free of primes?
I mean, $\forall \ k \in \mathbb{N}, \exists \ k$ consecutive positive integers none of which is a prime.
5
votes
3answers
90 views
Size of largest prime factor
It is well known and easy to prove that the smallest prime factor of an integer $n$ is at most equal to $\sqrt n$. What can be said about the largest prime factor of $n$, denoted by $P_1(n)$? In ...
1
vote
0answers
17 views
Using Gamma function to show the limiting case of Gordon's continued fraction as q approaches i.
A question similar to: How to derive the Golden mean by using properties of Gamma function?
The limiting case of Gordon's continued fraction when $q$ approaches $i$ yields:
$$\sqrt2 + 1 = ...
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
$$
\frac{2\cdot 3\cdot 7\cdot 8\cdot 12\cdot 13\cdots}{1\cdot 4\cdot 6\cdot 9\cdot 11\cdot ...
5
votes
1answer
100 views
Analytically continue a function with Euler product
I would like to estimate the main term of the integral
$$\frac{1}{2\pi i} \int_{(c)} L(s) \frac{x^s}{s} ds$$
where $c > 0$, $\displaystyle L(s) = \prod_p \left(1 + \frac{2}{p(p^s-1)}\right)$.
...
0
votes
1answer
42 views
Why does the theta function decay exponentially as $x \rightarrow \infty$?
I'm trying to understand the proof of the functional equation for the L-series of primitive, even Dirichlet characters.
For even, primitive characters we have $$\theta_\chi(x):=\sum_{n\in \mathbb{Z}} ...
5
votes
2answers
126 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
124 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 ...
2
votes
1answer
47 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
40 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 ...
4
votes
1answer
71 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
41 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
4
votes
1answer
50 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 ...
5
votes
5answers
176 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 ...
1
vote
0answers
21 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
47 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 ...
8
votes
2answers
100 views
Approximation of $\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$ [duplicate]
I am reading about the Riemann hypothesis, and the article mentioned the Li function:
$$\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$$
They said that this function can be approximated:
...
1
vote
0answers
68 views
By establishing a recurrence relation and using induction, or other-wise, show that this sequence is 3-adically Cauchy?
this is a question from a book I'm struggling with, please could you provide a clear proof
Consider the sequence of rational numbers
$a_1 = 1+3,a_2 = 1+\frac{3}{1+3},a_3= 1 + \cfrac{3}{1
...
2
votes
1answer
100 views
For what primes $p$ does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?
this is a question from a book I'm struggling with, please could you provide a clear proof
For what primes p does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?
kind thanks
0
votes
1answer
55 views
For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically
For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically and, when it does, to what limit?
0
votes
0answers
96 views
Show that the field of p-adic numbers is complete
this is a question from a book I'm struggling with, please could you provide a clear proof
Show that the field of p-adic numbers is complete
i.e. that a sequence of p-adic numbers converges if and ...
3
votes
1answer
43 views
$a_{[n/1]}+a_{[n/2]}+…+a_{[n/n]}=1$
The sequence $a_n$ satisfy
$$a_{[n/1]}+a_{[n/2]}+...+a_{[n/n]}=1,$$
for all $n \in \Bbb N$.
(the subscript $[n/k]$ is the integer part of $n/k$)
$Proof:$for any $k>0$,$$\lim_{n \rightarrow ...
4
votes
2answers
89 views
Using sum of logarithms of primes to prove the number of primes up to $n$ is $O(n/\log n)$
I need to show that the number of primes up to $n$ (i.e. $\pi(n)$) is $O(n/\log n)$.
In the previous exercise of this question I proved that ${\displaystyle \sum_{i=1}^{\pi(n)}\log p_{i}} \leq Cn$ for ...
3
votes
1answer
71 views
Understanding a very elementary property of factorials
I've seen this stated in a few places.
If $$\vartheta(x) = \sum_{p\le{x}} \log (p) \qquad \psi(x) = \sum_{m=1}^{\infty}\vartheta\left(\sqrt[m]{x}\right)$$ Then $$\log(x!) = \sum_{m=1}^{\infty} ...
2
votes
1answer
36 views
How to show $\sum_{d>x,P^+(d)\le y} \mu(d)/d = O (\log{y}\cdot e^{-\log{x}/2\log{y}})$?
I can't see why $$\sum_{d>x,P^+(d)\le y} \mu(d)/d = O (\log{y}\cdot e^{-\log{x}/2\log{y}})\;,$$
where $P^+(d)$ is the greatest prime factor of $d$.
Can anyone give a hint? Thanks.
4
votes
1answer
88 views
evaluate $\phi(50!)$
I want to evaluate $\phi(50!)$, where $\phi$ is the Euler totient function, so i take the factorization in primes of $50!$
$$2^{47}\times 3^{22}\times 5^{12}\times 7^8\times 11^4\times 13^3\times ...
2
votes
1answer
195 views
A convergence problem： splitting a double sum
I have been facing some difficulties with the following question.
For an absolutely convergent series $\sum_m a_m$, and the Mobius function $\mu(n)$, $x=(x_1,x_2)\in \mathbb{R}^2$, and $\alpha ...
5
votes
2answers
192 views
A cohomological statement equivalent to the Riemann Hypothesis
Is there a possibility for looking for a theory of cohomology and an equivalent cohomological statement for Riemann hypothesis over $\mathbb{Z}$?
5
votes
2answers
78 views
Bounds on a sum of gcd's
Does there exist a positive real number $C$ and a positive integer $M$ such that for all $n > M$ we have: $$\sum_{i=1}^n\sum_{j=1}^n\gcd (i, j)\ge Cn^2 \log n$$
This originally appeared as an ...
0
votes
1answer
67 views
What's the name of this class of transcendental numbers?
I'm considering the set $$\left\{\sin(k)\mid k\in\Bbb Z\backslash \left\{0\right\}\right\}.$$ All of its members are transcendental numbers, but together they don't form the complete set of all ...
3
votes
2answers
98 views
Sum of squares of sum of squares function $r_2(n)$
Let $r_2(n)$ denote the number of representations of $n$ as a sum of two squares.
What is known about the sum of squares of this function,
$\sum_{i=1}^n r_2(i)^2$
In particular is anything ...
6
votes
1answer
191 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 $\zeta(s)$ ...
1
vote
2answers
60 views
Basic Questions about the Properties of Mertens Function
I am astounded by how little information about Mertens function M(n) (partial sums of the Mobius function) is on the Internet. Thus, I would be thankful if someone could clear up some of my confusion. ...
6
votes
5answers
179 views
$(1+\frac{1}{n\log n})^n-1=O(\frac{1}{n})$.
When I solved a problem, I could solve it if I assumed that $(1+\frac{1}{n\log n})^n-1=O(\frac{1}{n})$
I tried to prove it, but I failed.
Actually, I don't convince if it is true.
Is it correct? If ...
