Questions on the use of the methods of real/complex analysis in the study of number theory.
8
votes
1answer
121 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
93 views
Possibly Wrong (Perhaps in the Right Direction Though)
I saw this question
Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $B ∈ \mathbb{Q}^m$. Suppose
that the system of linear equations $AX = B$ has a solution in
$\mathbb{R}^n$. Does it necessarily have ...
4
votes
1answer
28 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
64 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
$$
\begin{align*}
...
4
votes
1answer
213 views
Evaluating a series with the Möbius function and greatest common divisor.
Problem: Let $\gcd(a,b,c,d)$ refer to the largest integer $r$ such that $r$ divides each of $a,b,c,d$. Evaluate the series ...
6
votes
2answers
169 views
Proving two sequences identical
I found something quite interesting while browsing around the OEIS yesterday. I have no idea how to prove this (I don't even know if it's true in general, but Mathematica tells me that it holds up to ...
2
votes
2answers
52 views
Analytic method for number theory-do we have to assert second-order logic?
I am an undergraduate. I am just starting to study logic and analytic number theory at the same time, so please forgive me if I made an elementary misunderstanding.
A lot of theorem in number theory ...
4
votes
1answer
86 views
Binary vs. Ternary Goldbach Conjecture
Is there an "understandable" explanation of why the ternary Goldbach conjecture is tractable with current methods, while the binary Goldbach conjecture seems to be out of scope with current ...
3
votes
2answers
47 views
effective version of Mertens Theorem for the Euler product
I'm referring to the theorem given here, which is
$$\displaystyle\lim_{n\to \infty} \:\: \left(\frac1{\ln(n)} \cdot \left(\displaystyle\prod_{p\leq n} \frac1{1-\frac1p}\right)\right) \;\;\; = \;\;\; ...
1
vote
0answers
40 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
4answers
380 views
$\sum\limits_{d \mid n} \mu(d) \omega(n/d)=0$ for composite numbers. How?
I need some help with the last(?) step in a proof and I'm not sure how I should proceed... $\mu(n)$ is the Möbius function and $\omega(n)$ is the number of distinct prime factors. We see that for $n$ ...
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: $$\log x! = ...
0
votes
4answers
69 views
Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1} \frac{1}{p_n}$ convergent? [duplicate]
Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1}\frac{1}{p_n}$ convergent?
2
votes
1answer
57 views
Perron's formula (Passing a limit under the integral)
I want to understand why assuming that $\sum_{n \ge 1} \frac{a_n}{n^s}$ converges uniformly for $\mathrm{Re}(s) > \sigma > 0$ with $c > \sigma$ implies that
$$
\sum_{n \le x} \, \!\!^* a_n = ...
119
votes
4answers
4k views
Can you answer my son's fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?
My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question:
I am thinking of a number...
It ...
3
votes
1answer
155 views
Bounds on a sum involving the Möbius function
In Apostol's Analytic Number Theory, Apostol defines
$$A(x):= \sum_{n \leq x} \frac{\mu(n)}{n}$$
and proves that $A(x)=o(1)$ implies the Prime Number Theorem, by showing that
...
5
votes
2answers
123 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 $m$ ?
1
vote
0answers
42 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 $q$
$$\sum_{\rho=\beta+i \gamma: \Lambda(\rho, ...
4
votes
1answer
243 views
Divisor summatory function for squares
The Divisor summatory function is a function that is a sum over the divisor function.
$$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^u \lfloor\frac{x}{k}\rfloor - u^2, \;\;\text{with}\; u = \lfloor ...
5
votes
1answer
79 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 $$f(n) \ll ...
4
votes
1answer
235 views
Möbius function sum
If gcd(a,b)=1, $1\leq b\leq a$, and $\mu(k)$ is the Möbius function, what is $$\sum_{k=0}^\infty\frac{\mu(ak+b)}{(ak+b)^s}$$
Can it be expressed in terms of other functions? Can I get it in the form ...
5
votes
3answers
141 views
What is the set $\{x\in\Bbb R\mid \forall q\in\Bbb Q: q^x\in\Bbb Q\}$?
What is the set $\{x\in\Bbb R\mid \forall q\in\Bbb Q: q^x\in\Bbb Q\}$?
Of course $\Bbb Z$ is a subset of this set.
Are there any other? if not what is the proof? is there a good reference for it?
20
votes
5answers
811 views
What is so interesting about the zeroes of the $\zeta$ function
The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$
The ...
15
votes
5answers
771 views
Intervals that are free of primes
How can I prove that exists intervals as large as I want that are free of primes?
I mean, $\forall \ k \in \mathbb{N}, \exists \ k$ consecutive positive integers none of which is a prime.
5
votes
3answers
89 views
Size of largest prime factor
It is well known and easy to prove that the smallest prime factor of an integer $n$ is at most equal to $\sqrt n$. What can be said about the largest prime factor of $n$, denoted by $P_1(n)$? In ...
3
votes
1answer
120 views
How to derive the Golden mean by using properties of Gamma function?
The Golden mean known as $\frac{1+\sqrt{5}}{2}$.
How could one show the Golden mean can be expressed as
$$
\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
96 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
0answers
16 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:
$$\sqrt2 + 1 = ...
6
votes
1answer
122 views
Finding near-integers in a range
I have a (transcendental) constant $\alpha$ and a fixed parameter $\varepsilon>0.$ I'd like to find all positive integers $n<\ell$ for which $\|n\alpha\|<\varepsilon,$ where $\|x\|$ is the ...
0
votes
1answer
42 views
Why does the theta function decay exponentially as $x \rightarrow \infty$?
I'm trying to understand the proof of the functional equation for the L-series of primitive, even Dirichlet characters.
For even, primitive characters we have $$\theta_\chi(x):=\sum_{n\in \mathbb{Z}} ...
4
votes
1answer
69 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 ...
4
votes
2answers
87 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 ...
4
votes
1answer
95 views
Farey fractions in arithmetic progression
Let $\mathcal{F}_{Q;r,q}=\{\gamma=\frac{m}{n} | 0\leq m \leq n \leq Q, \gcd(m,n)=1, n \equiv r \mod q, \gcd(r,q)=1\}$. Usually, with no condition on arithmetic progression, then $\# \mathcal{F}_{Q}$ ...
2
votes
1answer
45 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 ...
13
votes
3answers
1k views
Non-increasing sequence of positive real numbers with prime index
If $a_n$ is a sequence of non-increasing positive numbers, then suppose we already know that
$$\sum_p a_p$$ converges, when $p$ runs over the primes, what should be used to prove that $$\sum_n ...
1
vote
2answers
41 views
showing that $\log(N) \leq \prod_{n \leq N} {(1-p^{-1})^{-1}}$
i can't see that $H_n \leq \prod_{n \leq N}{(1-p^{-1})^{-1}}$
and i can't see that $\log(N) \leq \prod_{n \leq N} {(1-p^{-1})^{-1}}$
p is prime and $H_n$ is harmonic series
4
votes
1answer
44 views
Clarkson's Proof of the Divergence of Reciprocal of Primes
In Tom Apostol's book, he credits the proof of the divergence of the sum of reciprocal of primes to Clarkson. To begin, we assume $\{p_n\}$ is an enumeration of the primes and ...
3
votes
2answers
238 views
Inverse of completely multiplicative function(s)
If $f$ is completely multiplicative, prove that
$$(f \cdot g)^{-1}=f \cdot g^{-1},$$
for every arithmetical function $g$ with $g(1) \neq 0$.
My main problem with this is that I can't even see why it ...
5
votes
5answers
172 views
Proving $\sqrt{2}\in\mathbb{Q_7}$?
Why does Hensel's lemma imply that $\sqrt{2}\in\mathbb{Q_7}$?
I understand Hensel's lemma, namely:
Let $f(x)$ be a polynomial with integer coefficients, and let $m$, $k$ be positive integers ...
1
vote
0answers
21 views
Existence of zeros of Mellin transform and properties of function to be transformed
Mellin transform of function $f(x)$ defined for $x\geqslant 0$ is given by
$$
f^\ast(z) =\int\limits_0^\infty x^{z} f(x) \frac{dx}{x}.
$$
I consider only exponentially decreasing (there exist such ...
2
votes
1answer
46 views
Stable points and the fundamental domain of the modular group
Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq 1/2\}$ its fundamental domain.
How ...
2
votes
1answer
189 views
A convergence problem： splitting a double sum
I have been facing some difficulties with the following question.
For an absolutely convergent series $\sum_m a_m$, and the Mobius function $\mu(n)$, $x=(x_1,x_2)\in \mathbb{R}^2$, and $\alpha ...
8
votes
2answers
100 views
Approximation of $\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$ [duplicate]
I am reading about the Riemann hypothesis, and the article mentioned the Li function:
$$\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$$
They said that this function can be approximated:
...
1
vote
0answers
68 views
By establishing a recurrence relation and using induction, or other-wise, show that this sequence is 3-adically Cauchy?
this is a question from a book I'm struggling with, please could you provide a clear proof
Consider the sequence of rational numbers
$a_1 = 1+3,a_2 = 1+\frac{3}{1+3},a_3= 1 + \cfrac{3}{1
...
3
votes
1answer
61 views
A strange quantum potential: $V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$
So I have a strange quantum potential I have been playing with:
$$V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$$
where $\mu$ is the Möbius function. This is what ...
7
votes
1answer
252 views
Is there a $k$ such that $a_n=\frac{n^k!}{(n^k!!)^2}$ converges?
Lately I have been playing around with the sequence $$a_n(k) := \frac{n^k!}{(n^k!!)^2}.$$
For $k=1$, it does not look much like it converges.
I don't know $k=2$ it converges, but it doesn't really ...
0
votes
0answers
88 views
Divergence of the Derivative of the Prime Counting Function
On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written
$$
\pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1}
$$
with $ \operatorname{R}(z) = ...
14
votes
1answer
384 views
Books about the Riemann Hypothesis
I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. Here is my list:
The Riemann Hypothesis: A Resource ...
2
votes
1answer
99 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
