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

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9
votes
3answers
80 views

Is there a monotonic $f$ such that $\sum f(n)$ diverges but $\sum f(p)$ converges?

(where the former summation is over natural numbers $n$ and the latter is over prime numbers $p$, and $f: \mathbb{N} \to \mathbb{R}$ is a monotonic function.) For the class of functions $f_s(n) = ...
3
votes
1answer
126 views

What is the value of $\sum_{p\le x} 1/p^2$?

My question is, what is the value of $$\sum_{p\le x} \frac{1}{p^2}?$$ More generally, what is the value of $$\sum_{p\le x} \frac{1}{p^n}?$$ How can we find it? For $\sum_{p\le x} 1/p$ the idea was ...
12
votes
1answer
162 views

Is $\sum\frac1{p^{1+ 1/p}}$ divergent?

Is $\displaystyle\sum\frac1{p^{1+ 1/p}}$ divergent? How can we prove that it is divergent or convergent in analytic number theory? I know what bound of the n-th prime number is, and that its order is ...
3
votes
1answer
71 views

Hardy-Ramanujan theorem's “purely elementary reasoning”

I'm reading through The normal number of prime factors of a number $n$. I'm confused by a remark on the second page: let $f(n)$ represent the number of distinct prime factors of $n$. Then we can ...
2
votes
1answer
34 views

How can one show that $\prod_{n<p\leq2n}p\leq C(2n,n)$?

I am trying to rove that $\prod_{n<p\leq2n}p \leq C(2n,n) \leq 2^{2n}$, where $C(2n,n)= \frac{2n!}{n! n!}$ and $p$ is prime. I can prove the second part by induction, but first part induction ...
1
vote
0answers
32 views

Is the integral $\int_1^{\infty} {A(t)}{t^{-s-1}}dt$ a holomorphic function of $s$?

My question is whether, for $Re(s)=\sigma > 3/4$, $$s\int_1^{\infty} \dfrac{A(t)}{t^{s+1}}dt$$ is holomorphic, where $A(x)=O(x^{3/4})$. Under absolute value, it is easy to see that the integral ...
1
vote
1answer
42 views

Number of the positive integers up to $\sqrt x $ generated by the prime factors of $x$.

Let $x$ be a natural number and $F_x$ be the set of distinct prime factors of $x$. One more let $\langle F_x \cup \{ 1 \} \rangle$ be multiplicative semigroup generated by the set. Then, the problem ...
2
votes
0answers
49 views

Find the integral values for which $\left(\pi(x+y)\right)^2=4\pi(x)\pi(y)$

Let $\pi(x)$ be the prime counting function. Find all integral values of $x,y$ such that, $$\left(\pi(x+y)\right)^2=4\pi(x)\pi(y)$$ I have no idea as to where to begin with. I think that probably ...
1
vote
0answers
28 views

Non-trivial odd characters mod m

I am stuck with this problem of marcus: I proved it when the charater is even. But I cannot prove the given formula when the character is odd. Please help.
1
vote
1answer
49 views

Can we define Mobius function for any real number and any complex number ?

All: To me, Mobius function is a bit mysterious. I just want to know if we can define Mobius function for any real number or any complex number ? Can anyone point out any resource on this ? Thank ...
3
votes
1answer
84 views

Absolutely convergent function

I am trying to show that if $\displaystyle\sum_{n\le x}f(n)=Cx+O(x^{3/4})$, where $f$ is non-negative multiplicative function and $C$ is a positive constant, then ...
0
votes
0answers
40 views

Using Perron's formula for asymptotic behaviors

I happen to read this post about trying to get the formula of $\sum_{n=1}^N n^m$ for Perron's formula. The general Perron's formula is $$\sum'_{n\le x} a(n)=\frac{1}{2\pi i}\int_{\text{Re ...
5
votes
1answer
85 views

Riemann Hypothesis, is this statement equivalent to Mertens function statement?

All: I saw one form of Riemann Hypothesis, it says: $$ \lim ∑(μ(n))/n^σ $$ Converges for all σ > ½ Is this statement same as the order of Mertens function is less than square root of n ?
0
votes
0answers
20 views

Estimates for a Mertens-type Product.

The first corollary of Theorem 8 of this paper by Rosser and Schoenfeld states that $$\prod_{p\leq x}\left(\frac{p}{p-1}\right)<e^{\gamma}(\log x)\left(1+\frac{1}{\log^2 x}\right)$$ for all $x\geq ...
2
votes
0answers
66 views

Riemann's explicit formula for $\pi(x)$

Riemann's explicit formula $J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2,$ where ...
1
vote
1answer
57 views

On the prime number theorem in shorts intervals

In 1988 Heath-Brown (" The number of primes in a short interval ", J. reine angew. Math. 389, 22-63) proved this theorem: Let $\varepsilon\left(x\right)\leq\frac{1}{12}$ be a non-negative function ...
3
votes
1answer
123 views

Cramer and Riemann Conjecture Implication

Cramer's conjecture gives $$p_{n+1}-p_n= O(\log^2 p_n)$$ while Riemann Hypothesis yields just $$p_{n+1}-p_n= O(\sqrt p_n\log^2 p_n).$$ Does Cramer conjecture on prime gaps imply Riemann Hypothesis ...
2
votes
0answers
32 views

parity problems for sieve methods, is it only for Selberg Sieve or for all sieve methods?

It is said that sieve methods have parity problems. Terence Tao gave this "rough" statement of the problem: "Parity problem. If A is a set whose elements are all products of an odd number of primes ...
1
vote
1answer
74 views

How to compute $\lim_{s \to 1} (s-1) \frac{\zeta'(s)}{\zeta(s)} $ ?

I wish to verify the conditions of a certain theorem to prove that the integral $$\int_{1}^{\infty} \frac{\psi(x) - x}{x^2} dx $$ converges. (Where $\psi(x) = \sum_{n\leq x} \Lambda (n) $, and ...
9
votes
2answers
263 views

What are some equivalent statements of (strong) Goldbach Conjecture?

What are some equivalent statements of (strong) Goldbach Conjecture ? We all know that Riemann Hypothesis has some interesting equivalent statements. My favorites are involved with Mertens ...
2
votes
2answers
175 views

Riemann Hypothesis and Prime Count

Let $\pi(a)$ be the number of primes below $a>0$. The prime number theorem states $\pi(a)\sim\frac{a}{\ln a}$. My question is trivial. Is $$\frac{a}{\ln a}\leq\pi(a)\leq\frac{a}{\ln a}+c\sqrt{a}\ln ...
2
votes
2answers
108 views

Chebyshev's first function prime count

How is Chebyshev's first function $$\vartheta(N)=\sum_{p\leq N}\log p$$ useful in counting primes? Can it alone be used to analytically derive the prime number theorem?
8
votes
2answers
193 views

Importance of the zero free region of Riemann zeta function

I have heard that for improving the error term in the Prime Number Theorem, we need better and better estimates on the zero free region. I have also heard that the best possible error term comes from ...
1
vote
0answers
51 views

Summing the reciprocal of $\phi(n)^2$

I am currently reading Vaughan's book 'The Hardy-Littlewood method' and am working on exercise 3.3 (page 37). I have tried working through a special case but that didn't help either. More concretely, ...
2
votes
1answer
37 views

How to find all Dirichlet characters

I want to know all the Dirichlet characters modulo $m$. I know that the number of such characters are $\phi(m)$. But how do find each and every character. for small moduli I could do it using some ...
2
votes
3answers
71 views

Arithmetic functions of particular type

Any there any natural functions real valued single variable that: changes (increases) values only at primes but otherwise stay constant (like a non periodic increasing staircase)? whose increase in ...
2
votes
0answers
71 views

Prove that $\sum\limits_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$. [closed]

Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that $$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$ ...
0
votes
1answer
46 views

Looking for methods for approximating an iterative equation regarding primes

In a previous question, I was looking for an equation for counting the number of the number of integers between $1$ and $x$ that have a prime factor besides $2$ or $3$. There were 2 iterative ...
5
votes
0answers
46 views

Kloosterman sum and multiples of 16

A Kloosterman sum is defined as $$K(a,b;m)=\sum_{0\leq x \leq m-1}_{\gcd(x,m)=1} e^{2\pi \mathcal{i} (ax+bx^*)/m}$$ where $a,b,m \in \mathbb{N}$ and $x^*$ is the inverse of $x$ modulo $m$. How can ...
1
vote
0answers
25 views

GCD of Arguments of Kloosterman Sum

A Kloosterman sum is defined as $$K(a,b;m)=\sum_{0\leq x \leq m-1}_{\gcd(x,m)=1} e^{2\pi \mathcal{i} (ax+bx^*)/m}$$ where $a,b,m \in \mathbb{N}$ and $x^*$ is the inverse of $x$ modulo $m$. It ...
2
votes
0answers
44 views

Riemann Zeta Function and Laurent Expansion

In the wikipedia page "1+2+3+4+..." http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF (and specifically in the section "Zeta Function Regularization")it is stated without reference ...
1
vote
1answer
155 views

Abscissa of convergence for a Dirichlet series

Let $\alpha \in \mathbb{Z}$ and $f(n) = n^{i \alpha n}$. What is the abscissa of convergence, $\sigma_c$, for the associated Dirichlet series, $\sum_{n=1}^{\infty} \frac{f(n)}{n^s}$? Since $|f(n)| = ...
0
votes
2answers
101 views

Prove that as $x\to\infty $, $\sum\limits_{p \leq x} \frac{1}{p \log \log p} \approx \log \log \log x$

Prove that as $x\to\infty$, $$\sum_{p \leq x} \frac{1}{p \log \log p} \approx \log \log \log x$$ Here sum is taken over primes.I tried to use the partial summation formula but could not ...
0
votes
1answer
49 views

Convergence of a certain series of Primes

This is a problem from Alan Baker's Comprehensive Course in Number Theory. We have to show that $\displaystyle \sum\limits_{p} \frac{1}{p (\log\log p)^{\delta}}$ converges for all $\delta >1$.Here ...
4
votes
1answer
48 views

equivalence between Chebyshev estimation and pi estimation in PNT

I searched, but though many posts are close, none of them are dublicates. Our version of PNT states that there is some $c$ s.t. $$\psi(x)=x+O(x\exp(-c\sqrt{\ln x}))$$ We have to prove equivalence ...
1
vote
2answers
99 views

References for Riemann Hypotheis giving the best bound for Prime Number Theorem

Which books cover the proof that Riemann Hypothesis is equivalent to the best error bound for the Prime Number Theorem? My understanding is that Riemann Hypothesis is equivalent to the best bound of ...
1
vote
0answers
57 views

Reference for Analytic Number Theory

Are there any good video lectures available on Analytic Number Theory? I have a decent background of complex analysis but I have just started Analytic Number Theory.
1
vote
1answer
52 views

Solving $x^n \equiv a \text{ (mod } p)$ in $\mathbb{Z}$

I want to show that for any integers $a$ and $n,$ ($n > 1$) there are infinitely many primes $p$ such that $$x^n \equiv a \text{ (mod } p).$$ When $n$ is odd, I used the fact that if $(a,p)=1$ ...
1
vote
1answer
29 views

Product involving Dirichlet characters $\prod_{i=1}^{\phi(m)}(1 - \frac{\chi_i(p)}{p^s}) =(1 - \frac{1}{p^{fs}})^{\frac{\phi(m)}{f}}$

While working with divisors in cyclotomic extensions of $\mathbb{Q},$ I came across this identity: Given a prime $p$ and an integer $m$ such that $(p,m) =1$, let $f$ be the smallest such the $p^f ...
0
votes
1answer
21 views

Nonvanishing of Dirichlet Series on Re(s)=1

I am reading a proof of Prime numbers in Dirichlet arithmetic progressions via this link: http://www.math.leidenuniv.nl/~evertse/ant14-7.pdf However, according to his lemma 7.6, the writer wanted to ...
3
votes
1answer
66 views

Mangoldt Lambda Sum Rearrangement (from proof of Logarithmic Derivative of Riemann zeta function)

Also, we have by the definition of Λ, $$\sum_{n\geq 1} \Lambda(n) n^{−s} = \sum_p(\log p) \sum_{n \geq 1}p^{−ns}$$ (From https://proofwiki.org/wiki/Logarithmic_Derivative_of_Riemann_Zeta_Function) ...
3
votes
1answer
96 views

Special case of prime number theorem for arithmetic progressions 4k+1

In terms of the proof of prime number theorem for arithmetic progressions, I have seen many proofs involving with the concept of "character". Is there an alternative way (without such a concept) to ...
3
votes
1answer
115 views

Dirichlet density

How to solve the following exercise: Let $q$ be prime. Show that the set of primes p for which $p \equiv 1\pmod q$ and $$2^{(p-1)/q} \equiv 1 \pmod p$$ has Dirichlet density ...
2
votes
0answers
43 views

Question on Dirichlet density

I did not understand the highlighted sentence of the exercise below: My question is: how does it follow that $f(x)=0$ has a solution mod $p$ implies that $f(x)$ (mod $p$) splits as the product of ...
1
vote
0answers
27 views

Specific question on dirichlet density

In a notes I found the following exercise and solution: I have a question. In the proof I admit the statement "the Dirichlet density of these prime ideals is $1/2$ " but i do not understand why the ...
1
vote
0answers
59 views

Logarithmic derivative and the riemann zeta function

I'm trying to prove the following theorem. Theorem (Zero free region): There exists $C>0$ such that $\sigma > 1-\frac{C}{log(\vert t \vert +4)}\Rightarrow \zeta(\sigma + i t)\neq 0$. In the ...
1
vote
0answers
17 views

bounds for, $|L_{\tau}(s)|$, a Dirichlet searies associated with Ramanujan tau function

The Dirichlet searies associated with Ramanujan tau function is defined as: \begin{equation} L_{\tau}(s)=\sum_{n=0}^{\infty}\frac{\tau(n)}{n^s}=\prod_{p \text{ ...
2
votes
1answer
48 views

Integrating Chebyshev theta function

I'm trying to compute the following integral ($ \vartheta(x) = \sum\limits_{p \leq x}\log(p) $) $$\int\limits_{0}^{\infty}\vartheta(e^x) e^{-(1+s)x} \text{dx}$$ The result is supposed to be $ ...
1
vote
0answers
32 views

Logarithm derivative of $ \zeta(s) $

I have just proved that $$- \frac{\zeta'(s)}{\zeta(s)} = \sum\limits_p\frac{\log(p)}{p^s - 1}$$ and am aiming to prove that $$ -\frac{\zeta'(s)}{\zeta(s)} = \sum\limits_p\frac{\log(p)}{p^s} + h(s) ...
4
votes
1answer
282 views

Primes in the sequences $1+3n$ and $1+4n$

I'm studying primes in two sequences. By analogy with the Chebychev's work, define the functions $$\psi_*(x)= \sum_{n\leq x} \Lambda(1+3n)\Lambda(1+4n)$$ and $$\theta_*(x) = \sum_{n\leq ...