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

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2
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0answers
32 views

Finite sequence created by reducing $n$ with each prime under $n$ ends in $0$?

Given $n$ a fixed integer we constuct the following sequence: $a_0=n$, $a_i=\lfloor \frac{a_{i-1}(p_i-1)}{p_i}\rfloor$. For what values of $n$ do we have $a_{\pi(n)}=0$? Computer calculation shows ...
2
votes
0answers
27 views

A linearly uniform but quadratically non-uniform set

I have been working on this problem for a while but have no clue at all. Fix a smooth cutoff function $\varphi: \mathbb R/\mathbb Z\rightarrow [0,1]$ supported on $[−\varepsilon-\delta, \...
0
votes
1answer
52 views

Bound a complex exponential sum when we can only estimate its argument

Suppose we have a function $f : \mathbb{N_0} \to \mathbb{R}$ for which we can give an estimate of its values, and say its values $f(n)$ are roughly uniformly distributed for $n$ in some range $[1,N]$. ...
1
vote
0answers
39 views

Meaning of “Form” in mathematics

There are a number of occasions in the mathematics where the word "form" is used: Modular form Bilinear form Quadratic form ... It seems that form is a special kind of function. But I cannot ...
1
vote
0answers
18 views

Can you discuss $\limsup_{n\to\infty}\frac{g_n}{\log^2p_n}\cdot\frac{\sigma(K_n)}{K_n\cdot\log\log K_n}$, where $g_n=p_{n+1}-p_n$ and $K_n\to\infty$?

Let $p_n$ the nth prime number, then we know that the nth gap is $g_n=p_{n+1}-p_n$. We define for $n>1$, $C_n$ as the set of integers such that $gcd(k,p_{n+1})=gcd(k,p_{n})=1$, this is $\{1\leq k &...
0
votes
0answers
37 views

Who extended the Euler Product Formula to all real $s>1$?

I believe Euler discovered this identity but only wrote them for particular values of $s$, then Chebychev extended to real $s>1$. However, I read in the book Riemann's Zeta Function, H.M. Edwards ...
0
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0answers
46 views

How do you justify these statements from $\psi(x)=x-\sum_{\rho:\zeta(\rho)=0}\frac{x^\rho}{\rho}-\log 2\pi-\frac{1}{2}\log(1-x^{-2})$?

In the second paragraph of page 343 of this paper, NOTICES OF THE AMS Vol. 50, No. 3 the authors wrote the following concerning to the sum $\sum_{n\leq x}\Lambda(n)$, where $\Lambda(n)$ is von Mangold ...
3
votes
1answer
52 views

Can you give an understandable strategy to compute the asymptotic behaviour of $\sum_{2\leq n\leq x}\frac{\Lambda(n)Li(n)}{n}$?

First I am looking a discussion about if the statements that I've deduced will find a good asymptotic formula, what of them you can discard as non useful,or one better than other, and ... second my ...
1
vote
1answer
46 views

Exploring the Dirichlet series of the sum of remainder function

I wolud like to learn and understand more some basic facts about Dirichlet series, for wich I want explore the following function, that is called the sum of remainders function, A004125 as Sloane's ...
0
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0answers
20 views

It is possible an application of Shapiro's Tauberian theorem for $\sum_{n\leq x}\frac{|M(n)|^{1+\alpha}}{\pi(n)^{1+\beta}}\left[\frac{x}{n}\right]$?

I would like to know if I can find some $\alpha,\beta\geq 0$ such that defining the sequence $a(1)=1$ and for $n>1$ as $$a(n)=\frac{|M(n)|^{1+\alpha}}{\pi(n)^{1+\beta}},$$ where $M(n)=\sum_{k\leq n}...
1
vote
1answer
64 views

Can you provide us an understandable and detailed explanation of the relationship between prime numbers and equidistibution theory?

In this site, in 2. the user that answered the question wrote "the prime number theorem is an equidistribution theorem, and the Riemann Hypothesis an optimal discrepancy bound. By bounding this ...
2
votes
1answer
56 views

Evaluating $\lim_{x\to\infty}\frac{1}{x}\int_2^x M(t)\cdot f'(t)dt$, where $M(x)$ is Mertens functions

Let $\mu(n)$ the Möbius function. I know that combining Abel summation formula, the Prime Number Theorem and l'Hôpital's rule I can deduce $$\lim_{x\to\infty}\frac{1}{x}\sum_{2\leq n\leq x}\mu(n)\cdot\...
0
votes
1answer
43 views

Vaughan's identity, a didactic example

I know that Vaughan's identity is one of the methods used in analityc number theory, I would like see an example, I say a simple example of application of this theorem for encourage to study the ...
0
votes
1answer
59 views

Poisson summation in analytic number theory, an example

I've read a theorem of a course of analytical methods in theory of numbers, and I want to ask to clarify it, since I know this theorem from other context. The theorem is Poisson summation formula (I ...
0
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0answers
34 views

Additive combinatorics: Switching $\mathbb{Z}/N\mathbb{Z}$ with $\mathbb{Z}$

For a positive integer $N$, denote $\mathbb{Z}_N = \mathbb{Z}/N\mathbb{Z}$. Now let $F$ be a field and let $f_1, \ldots, f_s : \mathbb{Z}_N \to F$. Can we find functions $\tilde{f_1}, \ldots, \tilde{...
1
vote
1answer
47 views

A didactic example of logarithmic measure

I've read that two mathematicians, in the recent past, studied limits as $X$ tends to infinite, like as $$\frac{1}{\log X}\sum_{x\leq X:\pi(x)<\int_{2}^x\frac{dt}{\log t}}\frac{1}{x},$$ where $\pi(...
1
vote
0answers
54 views

Can you estimate the difference of primes between numerator and denominator?

Let $p_n$ the nth twin prime, it is $p_n$ is a prime number and $2+p_n$ is also a prime. It is well know that Brun's theorem states (unconditionally) that $$\mathcal{B}=\sum_{n\geq 1}\left(\frac{1}{...
0
votes
0answers
22 views

Number of ways, modulo a prime $p$, to write $n$ as a sum $n = x_1^k + x_2^k + \cdots + x_s^k$

Removing the restriction on $p$, this is known as Waring's problem. The circle method has been used successfully to tackle this. Using analysis, nice estimates can be given. I wonder what analytic ...
1
vote
0answers
37 views

On an inequality involving primorial numbers.

Let $N_k$ denote the $k-th$ primorial number. That is, the product of the first $k$ primes and $\phi(n)$ be the Euler totient function. How can one show that there exists a constant $\theta>1$ ...
4
votes
1answer
33 views

What about $\sum_{\substack{2\leq n\leq y,\text{n prime}}}n\log\log n$ when $y=[x]\to\infty$?

For a real $x\geq 2$ and when we take $y= [x]$ its integer part, I am trying to study the asymptotic size or growth of $$\sum_{\substack{2\leq n\leq y,\text{n prime}}}n\log\log n,$$ I believe that ...
5
votes
1answer
132 views

Help to solve $\displaystyle \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} dz $

I need help in evaluating the following contour integral: $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} ds $$ It looks like a complicated ...
0
votes
1answer
41 views

Are there any known asymptotics for $\sum_{p\leq x} p$? [duplicate]

As a prospective undergraduate who has really benefited from his time on MSE thus far, i recently learnt that there exists asymptotic approximations for $\sum_{p\leq x} 1, \sum_{p\leq x} p, \sum_{p\...
2
votes
0answers
35 views

Question on a proof of the Euler product of zeta function

Let $\zeta(s)$ be the Riemann zeta function. Then we know it satisfies the Euler product for $\text{Re}(s) > 1$, $$ \zeta(s) = \prod_{p} (1 - p^{-s})^{-1}. $$ The proof I read, if I recall ...
2
votes
1answer
49 views

How are the nontrivial zeros of the Riemann zeta function calculated?

The Riemann zeta function, is the function of the complex variable $s$, defined in the half plane $\Re(s)>1$ by the absolutely convergent series $\zeta(s) = \sum_{n} n^{-s}$ and extends to the ...
0
votes
1answer
33 views

An upper bound for the Chebyshev function?

The Chebyshev functions are defined as $\psi(x) = \sum_{p^m \leq x} \log n$ and $\theta(x) = \sum_{p\leq x} \log p$, where $p$ is a prime, $m\geq 1$ is an integer and $n=p^m$ in $\psi(x)$. It is known ...
0
votes
1answer
35 views

Is the prime counting function differentiable?

Let $\pi(x)$ denote the number of primes not exceeding $x$. Is $\pi(x)$ differentiable ? My attempt: It is well known that $\log \zeta(s) = \int_{2}^{\infty} \dfrac{s\pi(x)}{x(x^s - x)} \mathrm d{x}$ ...
1
vote
1answer
44 views

estimation for n-th prime

The famous theorem of Hadamard and Vallee-Poussin https://en.wikipedia.org/wiki/Prime_number_theorem implies that $p_n\sim n\ln n$, so $C_1 n\ln n \le p_n \le C_2 n\ln n$ holds for all $n\ge 2$ with ...
0
votes
0answers
65 views

An asymptotic involving fractional parts

I guess this is quite well known, but I was not able to find the related result. I want to find an asymptotic estimate for the expression $\sum_{k=1} ^{C\lfloor L \rfloor} \sum_{n=1} ^{\infty} \...
1
vote
1answer
56 views

On the sum of the reciprocals of the zeros of $\zeta(s)$

It is well known that whenever $\rho$ is a nontrivial zero of the Riemann zeta function $\zeta(s)$, then $1-\rho$ is also a zero. But does the equality $\Re \sum_{\rho} \dfrac{1}{\rho} = \Re \sum_{\...
1
vote
1answer
33 views

Dense on the unit circle

I am reading: "It is sufficient to show that the points $z_n = e^{2\pi in \xi}$ $\:\:n = (1, 2, 3...)$ are dense on the unit circle. ( $\xi$ is an irrational number)" How is this possible? Can ...
0
votes
0answers
66 views

An analytic formula for the sum of the logs of primes.

I just read in Martin Klazar's Intoduction to Number Theory (page 53), that $\sum_{p\leq x} \log p - \log (p-1) = \log\log x + \gamma + O(1/\log x)$. Where $\gamma$ is the Euler-Mascheroni constant, $...
0
votes
1answer
37 views

On the sum of the logarithms of primes.

Let $p$ be a prime and $x$ be an integer. It is known that $\sum_{p\leq x} \log p = O(x)$, and i think this is equivalent to the Prime Number Theorem. ...
0
votes
1answer
40 views

What is an upper bound for number of prime powers and semi primes in the interval $[n^2+1,n^2+n]?$

What is an upper bound for number of prime powers in the interval $[n^2+1,n^2+n]?$ What is an upper bound for number of square free semi primes in this interval$?$
2
votes
1answer
80 views

Goldbach Conjecture, what are new research methods after Chen's work?

For Goldbach Conjecture, my understanding is that there are three major methods to attempt it: Schnirelmann density circle method sieve method (Chen used two parameter sieve method to get his ...
0
votes
1answer
148 views

What would the Riemann Hypothesis mean for the Prime Number Theorem?

The Prime Number Theorem states $\pi(n)\sim \dfrac{n}{\ln n}$. Would there be an equally simple expression if Riemann's Hypothesis were proved true? From Chebyshev Function, would $\pi(n)\sim \dfrac{...
4
votes
1answer
79 views

Probability that a number has $m$ indistinct factors

I just discovered Matlab's factor()-function, and I randomly typed in 20081294819, and to my surprise it only had two factors (5099 and 3938281)! I had expected many more factors for such a big number ...
0
votes
0answers
23 views

$p$-adic digits via character sums

Let $p$ be a prime and let $n = \sum_{k=0}^\infty n_k p^k$ be a $p$-adic integer with each $0 \leq n_k \leq p-1$. Fix $0 \leq c \leq p-1$. Is there a way to check whether the $i$-th digit $n_i$ equals ...
1
vote
1answer
78 views

Elementary proof of the prime number theorem?

The prime number theorem is equivalent to $\lim_{x \to \infty} \dfrac{1}{x} \left| \sum_{n\leq x} \mu(n) \right| = 0$, where $\mu(n)$ is the Mobius function. We know that $\left| \sum_{n\leq x} \mu(...
0
votes
1answer
56 views

On the log of the Riemann zeta function.

Let $\pi(x)$ denote the prime counting function. It is well known that $\log \zeta(s) = \int_{2}^{\infty} \dfrac{s\pi(x)}{x(x^s - x)} \mathrm d{x}$ where $\Re(s)\geq 2$. Inserting $s=4$, we have $\...
0
votes
1answer
35 views

Comparing Euler products

I have this $a(n)$ is unknown multiplicative function and $b(n)=n$. Let $\zeta(x)$ be Riemman zeta function. And $$B(x)=\zeta^2(x)A(x).$$ where $B(x)=\sum_{n\in \mathbb{N}}\frac{b(n)}{n^x}$ (same ...
3
votes
0answers
111 views

Combining Firoozbakht's conjecture and abc conjecture

Firoozbakht's conjecture states that for all $n\geq 1$ $$p_n^{\frac{1}{n}}>p_{n+1}^{\frac{1}{n+1}},$$ where $p_k$ the kth prime number. By asumption of this conjecture, for a fixed $n$, there is a ...
0
votes
0answers
27 views

What is Dirichlet class number formula for d when d is NOT a fundamental discriminant?

According to Wikipedia, Dirichlet published a proof of the class number formula for quadratic fields in 1839, but it was stated in the language of quadratic forms. Let d be a fundamental discriminant,...
2
votes
1answer
38 views

What are the equivalent statements of GRH using the Möbius or Liouville functions?

We all know that Riemann Hypothesis can be stated as properties of $\mu$ or $\lambda$, particularly in terms of the random behaviour of those functions with "square root" bounds. Are there similar ...
2
votes
1answer
55 views

What is an upper bound for number of semiprimes less than n?

A semi prime is a number which is product of two distinct prime number. What is an upper bound for number of numbers in the form pq less than n? $p,q$ are prime numbers smaller than $n$.
2
votes
1answer
71 views

Counting squarefree numbers which have $k$ prime factors?

How to find an asymptotic formula for this function below? $$f(n)=\sum_{pq\leq n}1$$ where $p$ and $q$ are different prime numbers. I guess we can write $$f(n)=\sum_{p\leq \sqrt{n}}\pi (\frac{n}{p})...
3
votes
2answers
95 views

What is an upper bound for number of prime powers less than $n$?

What is an upper bound for number of prime powers less than $n$? I mean the numbers in the form $a^b$ in which $b \ge 2$ and $a$ is a prime number. I have found that $\frac {\log n} {\log 2} + \frac ...
1
vote
1answer
40 views

Can class number $h(d)$ equal to zero for some $d$?

We know that $L(1, \chi)$ is related to the class number $h(d)$ with a constant. And this is one way that we can prove $L(1, \chi)$ not vanish on $s = 1$. What confused me is: we know that class ...
1
vote
2answers
62 views

an upper bound for number of primes in the interval $[n^2+n,n^2+2n]$

What is an upper bound for the number of primes in an interval of $n$ consecutive numbers? What is an upper bound for the number of primes in the interval $[n^2+n,n^2+2n]$?
3
votes
2answers
65 views

Convergence of prime zeta function for $\mathfrak R(s)=1$?

By doing some estimates for the partial sums of the Prime zeta function $P(s)=\sum_p p^{-s}$ for $\mathfrak R(s)=1$ I got that $P(1+i\alpha)$ converges for every $\alpha\neq0$... Since I did not ...
0
votes
0answers
23 views

dirichilet class number and non-vanish of L function at s = 1

All: I have been confused by dirichilet class number formula. We know that L ( 1 , χ ) is related to the class number h(d) with a constant. And this is one way that we can prove ...