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

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4
votes
2answers
278 views

How to proceed doing number theory?

I'm an undergrad majoring in mathematics. Being in first year I'm still exploring new branches of mathematics and till now, It is analysis and Number theory that I've come to have a great interest ...
5
votes
1answer
130 views

Bounding this arithmetic sum

I am interesting in bounding the arithmetic sum $$ \sum_{n \leq x} \frac{\mu(n)^2}{\varphi(n)}$$ (The motivation is that this is a sum that comes up a lot in sieving primes, in particular in the ...
1
vote
1answer
153 views

How to make Dirichlet character table modulo $5$

There are four reduced residue classes $\mod 5$, namely $1, 2, 3, 4$ and thus four Dirichlet characters $\mod 5$ since $\phi(5)=4$. I understand how to deduce that the character can be $1$ or ...
0
votes
2answers
92 views

Existing Algorithm / Code to calculate exact values of the Riemann Zeta function at even natural numbers?

I wanted to know if there's any existing algorithm to compute exact values of the Riemann Zeta function at even natural numbers? For example, it should compute $\zeta(4)$ as exactly $\frac{\pi^4}{90}$ ...
2
votes
2answers
77 views

How to determine growth rate of coefficients of generating function

For a given ordinary generating function $f(x)=a_0+a_1x+...$, are there any methods to determine the growth rate of its coefficients based on that of $f$ ? In particular if we are given the extra ...
0
votes
0answers
103 views

n-th harmonic number when n is prime

the $n$-th harmonic number $H\left(n\right)$ is usually defined by the sum $\sum_{k=1}^{n}\frac{1}{k}$. Now, we know there is no closed form for this number, however, in the Apostol "introduction to ...
7
votes
1answer
93 views

Volume of first cohomology of arithmetic complex

Let $K$ be a number field and consider the Arithmentic complex $\Gamma_{Ar}(1)^\bullet$ be defined by $$\begin{array} A\Bbb R^{r_1+r_2} & \stackrel{\Sigma}{\longrightarrow} & \Bbb R \\ ...
1
vote
1answer
63 views

Product of the logarithms of primes

I would like to know if there is a result for the product $$f(x)=\prod_{p\leq x}\log p,\quad \text{where $p$ is prime}.$$ A simple upper bound is $f(x)<(\log x)^{\pi(x)}$, where $\pi(\cdot)$ is ...
2
votes
1answer
121 views

How to prove the convergence of a series of prime numbers

I have a bit of a problem proving that the series: $$ \sum_{p\leq x} \frac{p\ln\left(p\right)}{x^2} $$ where the sum is extended over all prime numbers, converges to 0.5. Any ideas? Thanks in ...
0
votes
0answers
57 views

Specific form of integral representation of the Riemann zeta function

Is there an integral represenation of the Riemann zeta function of the form: $$\zeta(s) = f(s)+c\int_a^b\frac{g(x)}{x^{p(s)}}dx,$$ where $a,b,c\in\mathbb{R}$ with $a\neq b$, $p(s)$ is some ...
5
votes
2answers
114 views

$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$ for integer k

Can anyone compute $$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$$ for integer k? Can it be done in closed form, using only elementary functions, without the use of the Gamma function? For k=1, the closed ...
4
votes
0answers
75 views

Prime number theorem for Dedekind domains

Let $\mathscr P\subseteq \mathbb N$ be the set of prime numbers. The prime number theorem tells us that if $\pi(x)=|\{p\in\mathscr P\colon p\leq x\}|$ then $\pi(x)\sim \frac{x}{\log x}$. Now one could ...
2
votes
1answer
182 views

Integral representation of the Riemann zeta function

I've come across the following integral representation for the Riemann zeta function, $$\zeta(s) = \frac{s+1}{2(s-1)} + \frac{s}{8} - \frac{s(s+1)}{2\pi^2}\int_1^\infty \frac{(\tan^{-1}\cot(\pi ...
3
votes
1answer
52 views

Convergence of a series concerning the multiplicative order of 2

I was trying to bound the value of $v_p(2^n-1)$ and some of the series I obtain made me wonder about the following problem. Problem : When does the series $$\sum_{prime \: p} \frac{1}{(ord_p ...
5
votes
0answers
109 views

Are $ut + 1$ and $ut + t + 1$ both prime for some t for any $u$?

Conjecture : For any natural number $u$, there is a natural number $t$ such that $ut + 1$ and $ut + t + 1$ are both prime. So we get a solution of the equation $$au - b(u+1) = -1$$ with prime ...
3
votes
0answers
117 views

Ratio of maximal to minimal jump in the set of angle multiples (corrected)

(This is the corrected version of the question I asked here: Ratio of maximal to minimal jump in the set of angle multiples.) Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times ...
4
votes
1answer
130 views

Estimating integrals involving $\pi(x)$

While solving an exercise in analytic number theory, I ran into difficulty of estimating an integral of the form $\displaystyle\int_{1}^{x} \frac{\pi(t)}{t} dt$ where $\pi(x)$ is the prime counting ...
2
votes
0answers
71 views

Equivalence of three asymptotic statements

Question 22 of chapter 4 in Apostol's "Introduction to Analytic Number Theory" asks to show that the following three statements are equivalent: $$\psi(x) \log(x) + \sum_{n \le x} \Lambda(n) ...
1
vote
1answer
57 views

asymptotic of a product

So the question that I'm working on is the following. Show that $\Pi_{p\leq z}(1-\dfrac{1}{p})=\dfrac{C(1+\mathcal{o}(1))}{\log z}$. First off I take logs and just work with the sum and thisis what ...
4
votes
1answer
37 views

Integer solutions to an ellipsoid surface

Given the equation $$x^2+2y^2+5z^2+xz =n$$ where $n$ is any positive integer, what is the smallest odd integer for which no integer solution $(x,y,z)$ exists (i.e. $x,y,z$ are integers)? I know that ...
5
votes
2answers
98 views

Ways to calculate $\int_0^1 \frac{-\log x}{1+x}\ \mathrm dx$

I came across the integral $$ \int_0^1 \frac{-\log x}{1+x}\ \mathrm dx = \frac{\pi^2}{12}, $$ which can be calculated as $\frac 1 2 \zeta(2)$ using analytic number theory. I'm interested if this ...
8
votes
3answers
282 views

Counting the Number of Integral Solutions to $x^2+dy^2 = n$

It is a well known result that the number of integer solutions $(x,y), x>0, y\ge 0$ to $x^2+y^2 = n$ is $\sum_{d|n}\chi(d)$, where $\chi$ is the nontrivial Dirichlet character modulo $4$ such that ...
1
vote
2answers
77 views

Function approximating this product

Is there any function approximating, for large values of $p$, the quotient between the product of all primes and the product of all primes $-1$? Basically: $2/1 \cdot 3/2 \cdot 5/4 \cdot 7/6 \cdot ...
4
votes
1answer
86 views

Euclid's method to estimate $\pi(x)$ prime numbers

The Euclid's method to prove that there are infinitely many primes goes as if $p_1,\dots,p_n$ are all the primes, then $p_1\dots p_n+1$ must have a prime divisor which is not among $p_1,\dots,p_n$. ...
7
votes
1answer
155 views

Is the Green-Tao theorem a consequence of the Euler's theorem?

The Erdős-Turán conjecture states that If $A\subset\mathbb{N}$ is such that $$ \sum_{n\in A} \frac{1}{n} = \infty,$$ then $A$ contains arithmetic progressions of any given length. I'm ...
1
vote
2answers
122 views

Does dividing by zero ever make sense? [duplicate]

Good afternoon, The square root of $-1$, AKA $i$, seemed a crazy number allowing contradictions as $1=-1$ by the usual rules of the real numbers. However, it proved to be useful and ...
0
votes
0answers
130 views

An identity about Dirichlet $\eta$ Function

We know the Dirichlet $\eta$-function is defined as the analytic continuation of $$\eta(s) = \sum_{i=1}^\infty \frac{(-1)^{n-1}}{n^s} \quad \Re(s)>0$$ I find an identity for the values of this ...
2
votes
0answers
68 views

Liouville function and PNT

The Big Omega function is defined as the number on non-distinct prime factors of an integer. I.e. $\Omega (2^a3^b...p^z)=a+b+...+z$, and the Liouville function is defined as ...
0
votes
1answer
36 views

Sieve dimension of union of two sets.

Let $P$ be a set of primes $\leq p$. Let $A$ be a set of all integers $\leq x$ in which the elements in $A$ would avoid two classes mod $p_i$ for all $p_i \leq p$ (except $2$,$3$). My understanding ...
5
votes
1answer
116 views

On a constant defined by Ramanujan.

In the second letter to Hardy Ramanujan writes about the number of prime numbers less than $n$ there he writes. Here this constant $\mu$ facinated me . What is its closed form? and How to compute ...
0
votes
0answers
38 views

Is the explicit formula for the second chebyshev function unique?

Is the explicit formula for the second chebyshev function unique ? Or is it possible there are multiple explicit formula ? Are there explicit formula's given as an infinite product over the zero's ...
2
votes
1answer
67 views

What is the sum of this series? Dirichlet $L$-Function

$$\sum_{n>0} \frac{\mu(n)}{n^s}$$ Sum from 1 to infinity of The Möbius function$/n^s$, i.e., Möbius function/Riemman-zeta function? Sorry, I forgot to mention that the way that I am suppose to ...
5
votes
2answers
195 views

Solving an integral coming from Perron's formula

In analytic number theory, Perron's formula says that $$ \sum_{1 \leq k < n} a_k + \frac{1}{2}a_n = \int_{c - i\infty}^{c+i\infty} f(s)\frac{n^s}{s}ds, $$ where $f(s) = \sum_{k \geq 1} a_k/k^s$ ...
7
votes
1answer
123 views

Euler totient variation identity

This is problem 11 part b in chapter 3 of Tom M. Apostol's "Introduction to Analytic Number Theory". A variation on Euler's totient function is defined as $$\varphi_1(n) = n \sum_{d \mid n} ...
0
votes
1answer
101 views

On Euler totient function sum

Let $q$ an arbitrary integer. Is there any chance of getting a bound like $$\underset{d\mid q}{\sum}\frac{1}{\phi\left(q/d\right)^{2}}\ll\frac{1}{\phi\left(q\right)^{2}}?$$
2
votes
2answers
91 views

Prime number theorem and how many primes are close to $x$ for sufficiently large $n$

The prime number theorem states: $$ \lim_{x-> \infty}{\frac{\pi(x)}{\frac{x}{ln(x)}}} = 1 $$ I was trying to get a better understanding on the intuition on that statement and more importantly, I ...
1
vote
0answers
66 views

Estimation of a logarithmic sum

I need to estimate the sum $$ \underset{r=2}{\overset{t}{\sum}}\left(\frac{\log\log r}{r}\right)^{2}. $$ I tried to use the Abel's partial summation, and I got $$ \frac{(\log\log ...
0
votes
0answers
25 views

How can I prove the functional equation of L-function?

Let $L(f, s)$ be a L-function with conductor $q( f )$ and gamma factor $\gamma ( f, s )$. Let $\Lambda \left( f, s \right)$ be a complete L-function, i.e, $$\Lambda \left( f, s \right) = q \left( f ...
2
votes
1answer
56 views

Determination of all prime numbers which give integer solution of a particular summation.

Determine all primes numbers $p$ such that $$p \sum_{k=0}^{n}\frac{1}{2k+1} \in N$$ for a given positive number $n$
3
votes
1answer
101 views

Bound of the sum $\sum_{p\le n}\frac{1}{\log(p)}$

While doing a sum I came to the sum $\displaystyle\sum_{p\le n}\dfrac{1}{\log(p)}$. Where the $\log$ is the natural logarithm. It was easy to prove that $\displaystyle\sum_{p\le ...
1
vote
1answer
48 views

on the generating series $\sum_{n\geq 1}\frac{\sigma_a(n)}{n}x^n$

I was reading about the divisor function on Wikipedia, and I stumbled upon the formula $$\sum_{n\geq 1}\frac{\sigma_a(n)}{n^s}=\zeta(s)\zeta(s-a).$$ Here $\sigma_a(n)=\sum_{d|n}d^a$ for an integer ...
10
votes
1answer
498 views

Values of hypergeometric functions

Let $_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;c)$ denote the generalized hypergeometric function. Let $A \subset \mathbb R$ be the set of all values of $\ _pF_q(\cdot)$ at rational points $a_i,b_j,c\in ...
8
votes
2answers
327 views

How often is a sum of $k$ consecutive primes also prime?

Let's define a $k$-sum as a sum of $k$ consecutive primes. For example, $15=3+5+7$ is a $3$-sum. How many $k$-sums are themselves prime? Here's one way to formulate the question more precisely: What ...
4
votes
1answer
58 views

Linnik's theorem for kth prime in the residue class

Linnik's theorm says that for any modulus $m$, the smallest prime in a given residue class mod $m$ cannot be too large: $$ p(a,m)\ll m^L. $$ where $L$ is a constant which has been improved by many ...
16
votes
1answer
263 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
votes
1answer
72 views

Properties of Arithmetic Functions

I was recently working on arithmetic functions and using Perron's formula to obtain asymptotic estimates. One observation I made was that the Dirichlet series often can be written in terms of the ...
5
votes
2answers
230 views

Riemann Hypothesis and the prime counting function

This article on the prime counting function mentions that the Riemann Hypothesis is equivalent to the statement $$|\pi(x)-\rm {li}(x)|\le \frac {1}{8\pi}\sqrt {x}\log (x)\text { for all }x \geq 2657 ...
2
votes
1answer
41 views

Asymptotics for $p$-series with $p=1/2$

Reading solutions to a practice exam, and I come across this: $$ O\left(\sum_{d \leq \sqrt{x}} {1 \over \sqrt{d}}\right) = O\left(x^{1/4}\right). $$ There are $O(\sqrt{x})$ terms in the sum, which ...
-1
votes
1answer
63 views

Sequence of numbers with a special property [closed]

Prove that the sequence a(n) = 2013 + 317n, where n is any nonnegative integer, generates infinitely many palindromic numbers.
1
vote
1answer
205 views

Analytic number theory books after Apostol

I am planning to learn some classical results on analytic number theory. I have read Apostol's Introduction to Analytic Number Theory, but nothing about algebraic number theory. Can anyone recommend ...