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

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52 views

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

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 $$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$ ...
4
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1answer
38 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 ...
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1answer
26 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
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4answers
4k views

What is the analytic continuation of the Riemann Zeta Function

I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation. What is this other ...
5
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0answers
37 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 ...
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2answers
40 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 ...
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0answers
22 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 ...
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2answers
67 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 ...
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0answers
29 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 ...
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0answers
17 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)| = ...
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2answers
74 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
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0answers
29 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.
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1answer
40 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 $(a,p)=1$ ...
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1answer
23 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 ...
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1answer
16 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
79 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 ...
3
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1answer
71 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
35 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) ...
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0answers
33 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
18 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 ...
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36 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 ...
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1answer
248 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 ...
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0answers
44 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
14 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{ ...
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1answer
19 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 $ ...
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0answers
30 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) ...
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1answer
19 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)} ...
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1answer
24 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} ...
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1answer
26 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 ...
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1answer
44 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 ...
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3answers
47 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 ...
2
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0answers
45 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
39 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 ...
26
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1answer
2k 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 ...
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0answers
24 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 ...
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1answer
33 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: ...
4
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1answer
74 views

Sign of Ramanujan $\tau$ function

The Ramanujan $\tau(n)$ seemed to have random positive/negative signs: ...
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0answers
166 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 ...
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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 ...
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1answer
685 views

Are there infinite many $n\in\mathbb N$ such that $\pi(n)=\sum_{p\leq\sqrt n}p$?

Are there infinite many $n\in\mathbb N$ such that $$\pi(n)=\sum_{p\leq\sqrt n}p,\tag{1}$$ where $\pi(n)$ is the Prime-counting_function? For example, ...
3
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1answer
49 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 ...
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2answers
392 views

Residue of Rankin-Selberg L-function for non-trivial nebentypus

Let $f\in S_k(\Gamma_0(N),\chi)$ be a normalized holomorphic newform (i.e. weight $k$, level $N$, nebentypus $\chi$) and write its Fourier expansion as $$ f(z)=\sum_{n\ge 1} ...
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0answers
32 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 ...
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0answers
96 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 $\tau(n)=\sum_{d|n}1$ ...
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0answers
29 views

Modular Equation and Modular Forms?

I'm reading Ramanujan's Notebook now and I see some kinds of "Modular Equation". At first I think Modular Equation is just a set of interesting fomulaes, but wikipedia says that Modular Equation is ...
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0answers
23 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} ...
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1answer
33 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
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1answer
31 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
votes
3answers
62 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 ...
7
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1answer
138 views

An Inequality Invollving The Riemann Zeta Function

I'm having trouble proving the following inequality for $2<r<3$: $$(1+2^{-r})\frac{(3^r+1)^2}{3^{2r}+1}>\frac{\zeta(r)}{\zeta(2r)}.$$ I can easily plot the graph, and the inequality clearly ...