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

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4
votes
1answer
113 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 ...
0
votes
1answer
69 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
66 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
52 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
92 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
91 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
54 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
115 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
51 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
90 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
51 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
102 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 ...
2
votes
0answers
34 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
99 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
112 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
106 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
56 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
54 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
33 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 ...
6
votes
2answers
86 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 ...
7
votes
0answers
181 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
69 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
77 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
124 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 ...
0
votes
2answers
90 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
84 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 ...
1
vote
0answers
55 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
25 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
75 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
31 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
52 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
137 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 $f(s) = \sum_{k \geq 1} a_k/k^s$ ...
6
votes
1answer
107 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
81 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
80 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
49 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
21 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
53 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 $n$
2
votes
1answer
76 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
44 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
370 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
287 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
50 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 ...
11
votes
0answers
157 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, ...
2
votes
1answer
64 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
156 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
40 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
60 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
127 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 ...
4
votes
3answers
122 views

What is the best way to supplement a complex variables class to make it more complete for a math major?

For the upcoming semester I plan on a taking a “complex variables” course that many people, including myself, would not consider a true complex analysis class. I know that the course will likely use a ...