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

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1answer
45 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
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1answer
26 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
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1answer
20 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
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1answer
53 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
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1answer
84 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
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1answer
107 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
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0answers
37 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 ...
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0answers
20 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 ...
0
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0answers
50 views

any online video course for analytic number theory or elementary number theory?

All: I am looking for online video course on analytic number theory for self-study. On Youtube, there are a few seminars, but no complete course for a semester or a year. Can anyone point out if ...
1
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0answers
56 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 ...
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0answers
15 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
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1answer
28 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
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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
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1answer
259 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 ...
1
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1answer
25 views

Amount of numbers that are coprime to a Mersenne number

Let $M_p = 2^p-1$ be a Mersenne number, where $p$ is prime. Is it known that almost every number in the interval $[1, M_p]$ is coprime to $M_p$? That is, is it known that $$ \lim_{p \to \infty} ...
2
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3answers
53 views

Probability that a Mersenne number is prime

Let $p$ be a prime and let $M_p = 2^p-1$ be a (Mersenne) number. Is there any known result on the probability that $M_p$ is prime? In particular is it known whether the probability tends to $1$ as $p ...
0
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1answer
48 views

How to prove that $\omega (n) = O\Big{(} \frac{\log(n)}{\log(\log(n))}\Big{)}$ as $n \to \infty$?

Let $$\omega(n) := \text{number of distinct primes dividing } n. $$ How can one prove that $\omega (n) = O\Big{(} \frac{\log(n)}{\log(\log(n))}\Big{)}$ as $n \to \infty$? I know that $\omega(n)! \leq ...
2
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0answers
46 views

symmetrized partial sums for $\zeta(s)$ and $\eta(s)$ in the critical strip

$\def\Re{\operatorname{Re}}$ We start with $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}\qquad \Re(s)>1\tag{1}$$ $$\zeta(1-s)=\sum_{n=1}^{\infty}\frac{1}{n^{1-s}}\qquad \Re(s)<0\tag{2}$$ ...
3
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1answer
42 views

evaluate two sums in analytic number theory

How should I evaluate the following sums 1, $\sum_{p\leq t}\frac{log^2(p)}{p}$ where the sum is taken over all prime numbers. 2, $\sum_{n\leq X}\frac{\Lambda^2(n)}{n}$ where $\Lambda(\cdot)$ is ...
0
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1answer
29 views

How to answer the following question regarding a certain number of primes in a certain interval?

For an analytic number theory homework assignment, we are asked to prove the following (using the Prime number theorem $\pi(x) \sim x/\log(x)$ as $x \to \infty$ ): For every $\epsilon > 0 $ and ...
2
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0answers
32 views

Largest prime factor of a Mersenne number with exactly two prime divisors

For a prime $p$, let $M_p = 2^p-1$ be a (Mersenne) number with exactly two prime divisors, and let $P(p)$ be the largest of these two. Clearly $P(p) > \sqrt{M_p}$. This is very likely a hard ...
1
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1answer
34 views

a possible period of 124 for the sign of Ramanujan $\tau(3^n)$ function

The Ramanujan $\tau(n)$ seemed to have random positive/negative signs: ...
0
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1answer
20 views

Proving a certain function involving the Riemann-Zeta function is non-increasing

Show that $ f(x) = \frac{\zeta(x -2)}{\zeta(x-1)} \qquad x > 3, $ where $\zeta$ is the Riemann-Zeta function, is non-increasing. My attempt was to use $\zeta(s) = \frac{1}{\Gamma(s)} ...
4
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1answer
79 views

Sign of Ramanujan $\tau$ function

The Ramanujan $\tau(n)$ seemed to have random positive/negative signs: ...
2
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1answer
35 views

Can we tell if a number is prime by the number of its partition ?

Can we tell if a number is prime by the number of its partition ? Or in general, how much can we know about a number itself from its partition function ? I understand that Ramanujan has some ...
7
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0answers
172 views

Number of ways to express a binary number in a certain way

So I'm working on a problem where I get to a point where I have to count the number of solutions to an equation or at least find a decent upper bound to be used in an estimate I need later. The ...
3
votes
1answer
54 views

Determining the asymptotics of the Summatory function of an Arithmetic Function

We define the arithmetic function: $\displaystyle f(n) = \max\limits_{p^{\alpha} || n} \alpha$, that is if $\displaystyle n = p_1^{\alpha_1}\cdots p_k^{\alpha_k}$ (prime factorization of $n$) then ...
0
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0answers
38 views

Koch's version of the Riemann hypothesis for $x=p^2$

According to von Koch 1991, if the Riemann hypothesis is true, then the for the prime counting function $$\pi(x)=Li(x)+O(\sqrt x \log x)$$ For this equation, does there exist any reference or does ...
3
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0answers
103 views

Divisor summatory function for squares plus one

As an exercise for my Analytic Number Theory course, I need to prove, using Dirichlet hyperbola method, that: $\sum_{n\leq x}\tau(n² + 1)= {3\over\pi}x\log(x) + O( x)$, where $\tau(n)=\sum_{d|n}1$ ...
0
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0answers
26 views

How to prove that $\zeta(s)<0$ for $s \in (0,1)$ using a particular expression for the Riemann zeta function?

Like in this question, I would like to prove that $\zeta(s)<0$ for $s \in (0,1)$. However, I have to use the expression $$\zeta(s) = \frac{1}{s-1} + 1 -s \int_{1}^{\infty} ...
1
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1answer
34 views

Upbounds for Ramanujan $\tau(n)$ function

For prime $p$, $|\tau(p)|\le 2p^{11/2}$. I am looking for the upbound for $|\tau(n)|$,$n\in \mathbb{N}$. Thanks- mike
1
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1answer
35 views

express the dirichlet series for the sequence d(n)^2 in terms of riemann zeta.

Prove that $$\sum_{n=1}^\infty d(n)^2n^{-s}=\zeta(s)^4/\zeta(2s)$$ for $\sigma>1$ what i did: I already proved this formally, that is, without considering convergence. I use euler products, ...
3
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3answers
66 views

How to prove that $ \lim_{u \downarrow 1} (u-1) \zeta(u) =1 $?

I would like to prove that $$ \lim_{u \downarrow 1} (u-1) \zeta(u) =1 \quad .$$ However, I am not sure which form of the Riemann-zeta function I ought to pick in order to compute this limit. I ...
4
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1answer
40 views

Existence of primes $q$ such that $p\mid q-1$ and $q\mid p^n-1$

Let $p$ be a prime. By Dirichlet's theorem on arithmetic progressions, there are infinitely many primes $q$ such that $p\mid q-1$. Must there be also primes $q$ such that $p\mid q-1$, and also, $q ...
2
votes
2answers
45 views

Can sieve method prove ternary (three) prime Goldbach conjecture?

Can sieve method prove ternary (three) prime Goldbach conjecture (Vinogradov Theorem) ? I had done some research, I could not find any articles on this. Can anyone provide some help on this ? I ...
0
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0answers
19 views

what is the best Schnirelmann Constants?

what is the best Schnirelmann Constant for Goldbach Conjecture ? On http://mathworld.wolfram.com/SchnirelmannConstant.html the best Schnirelmann Constant is 7 ( from Ramaré ) My understanding is ...
2
votes
1answer
92 views

Corollaries of Green-Tao Theorem?

there is already a good thread which discusses some corollaries of the Green-Tao Theorem, here: Constructing arithmetic progressions The question I was wondering about is of a similar flavor but ...
0
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1answer
24 views

Using Moebius Inversion to solve a functional equation.

I am reading about Moebius inversions, and I am given the following claim: $\sum_{d=1}^{\infty} f(z^d) = g(z)$ with $g(z) = O(z)$ at $z=0$, implies that $f(z) = \sum_{d=1}^{\infty} \mu(d)g(z^d)$ by ...
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0answers
52 views

Any heuristic explanation on why sieve methods can not prove Goldbach conjecture?

Any heuristic explanation on why sieve methods can not prove strong Goldbach conjecture ? I remember that Terence Tao published a blog and gave an heuristic explanation on why circle methods very ...
0
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1answer
56 views

Proving Euler Summation by Parts Without Using Integration by Parts

Assume $f$ has continuous derivative $f'$ on [a,b]. Prove the following summation formula, without using partial integration: \begin{equation} \sum_{a< x \le ...
3
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1answer
30 views

Proving that $(2 \pi i)^{-1} \int e(\pi v^2/y^2) x^v y^{-1} dv = e(-\pi (\log x)^2 y^2 /4)$

I've seen a particular integral transform (an inverse Mellin Transform) used a few times, but I don't know how it's proved. In particular, I'm trying to prove $$\frac{1}{2\pi i} \int_{(2)} e^{\pi ...
0
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0answers
39 views

Definition of Hecke Operator on modular form of half-integral weight

We define, for $f\in M_{k/2}(\tilde\Gamma_1(N))$, $T_{p^2}f :=p^{k/2-2} f|[\tilde\Gamma_1(N)\zeta_{p^2}\tilde\Gamma_1(N)]_{k/2}$, where $\zeta_{p^2}$ is the lift of $\alpha=\begin{pmatrix} 1 & ...
2
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0answers
50 views

What is the relationship between GRH and Goldbach Conjecture?

We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, ...
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0answers
21 views

Please recommend the most easy to read analytic number theory book for self study [duplicate]

All: Can anyone recommend the most easy to read analytic number theory book for self study ? Prefer with hint to exercise. I have Apostol's, Introduction to Analytic Number Theory, just want to see ...
0
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1answer
24 views

Finite measure on positive integers

Disclaimer: I am sure that this idea is not at all new, but I have had trouble locating content directly related. I humbly accept that this question may be the result of a brain fart. Suppose that ...
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2answers
41 views

Question about Mobius function.

Let $N \in \mathbb{N}.$ I would know if is it true that $$-\underset{k\mid N}{\sum}\mu\left(k\right)\log\left(k\right)>0.$$I know that $$-\mu\left(k\right)\log\left(k\right)=\underset{r\mid ...
10
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3answers
197 views

likely open number theory problem: finite sum of $\zeta(2)$ equal to a square of rationals

Which $n$ can let $S=1+\frac14+\frac19+\cdots+\frac1{n^2}$ be a square of a rational number? Obviously, $1$ and $3$ work, but how to prove they are the only ones? I think this problem is really hard. ...
2
votes
1answer
57 views

Periodicity over the prime indices of a multiplicative sequence implies periodicity?

I have a real sequence $(a_p)$ indexed by the prime numbers which takes values -1, 0, or 1, having the property that $a_p=a_q$ whenever $p\equiv q \pmod m$, where $m$ is a fixed integer $>2$. I'm ...
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1answer
62 views

any computational analytic number theory book?

All: Can anyone recommend an introduction computational analytic number theory book ? I am mainly interested in using computer software to verify and confirm analytic number theorem, things related: ...
5
votes
1answer
110 views

How many numbers are products of $p^p$?

Consider the set $\mathcal{S}=\{1,4,16,27,\ldots\}$ of numbers which are products of numbers of the form $p^p$ for $p$ prime. ($\mathcal{S}$ is A072873 in the OEIS.) Note that multiple primes are ...