Questions tagged [analytic-number-theory]

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

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18 views

Condition for ({$n\alpha$},{$n\beta$}) being dense on [0,1]$\times$[0,1] [duplicate]

What is the condition for ({$n\alpha$},{$n\beta$}) being dense in [0,1]$\times$[0,1]? (Here {a} represents the fraction part of a). If the qustion changes to ({$n\alpha$},{$m\beta$}), then it is ...
4
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0answers
131 views

Inequation in paper from Terence Tao on the Collatz Conjecture

I try to understand Terence Tao's paper on the Collatz Conjecture [1909.03562], but got stuck on page 25. We have $n$ copies of a geometric random variable of mean $2$, denoted by $a_i$ and $a_{[i,j]}$...
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0answers
40 views

Why is the following equation true?

This is a fairly simple question but it seems really confusing for me. So basically I am reading the proof of the composition formula of Hecke operators: $$T(m)T(n)=\sum_{d\mid (m,n)}d^{k-1}T(mn/d^2)$$...
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0answers
43 views

Why should $\sum_{m=1}^N e(\alpha m^3)$ be big for some $\alpha?$

I'm going through a "circle method" proof of the fact that every large enough natural number $n$ is the sum of nine cubes. At some point a lot of control over the function $$f(\alpha)=\sum_{...
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17 views

Why does all entire cusp form can be written as $\Delta h$ where $h$ is an entire modular form of weight $k-12$?

I was going over the proof that all entire modular forms are expressible as polynomials of Eisenstein series $G_4$ and $G_6$. The proof is supported by inducting on the weight of modular forms. And an ...
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0answers
44 views

Exponential of big O

In Hildebrand's lecture notes on analytic number theory, I have that $$\sum_{n\leq N} \log n = N(\log N -1) + 1/2\log N + c + O(1/N)$$ And notes directly jumps from here to this $$n! = C\sqrt{n}n^ne^{-...
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2answers
77 views

Are there infinitely many primes which are the sum of the two sides of a right triangle?

Of the first $9.4 \times 10^9$ primitive Pythagorean triplets (generated using $s^2 - r^2, 2rs, s^2 + s^2$), nearly $16.5\%$ are such that the sum of the two orthogonal sides is a primes i.e. $s^2 - r^...
2
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1answer
82 views

Do you know a name and expression of this function of Lehmer?

In the paper "A new calculus of numerical functions." (D.H. Lehmer, 1931) Lehmer defines a function d as follows. The function d(i,n) may be defined as follows: d(i,n)=0, if i is not a ...
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47 views

Residues of the (modern) Approximate Functional Equation

This is adapted from 'Analytic Number Theory' by Iwaniec and Kowalski. I think the result is off by a sign but I am not sure. Let G(z) be an even holomorphic function bounded on $|Re(z)| < 4$ with $...
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1answer
54 views

Prove asymptotic equivalence of $\text{li}(n)$ and $n/\ln(n)$

The prime number theorem, PNT, states that the prime counting function $\pi(n)$ is asymptotically equivalent to Gauss' first approximation: $$\pi(n) \sim \frac{n}{\ln(n)}$$ We know this means that $$\...
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32 views

Triangular numbers in arithmetic progressions?

Does anybody know if there is an equivalent to the Siegel-Walfisz theorem (or prime number theorem) for triangular numbers in arithmetic progressions? The Siegel-Walfisz theorem essentially tells us ...
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0answers
63 views

Show that $\sum_{m\leq x , (m,d)=1}\frac{\mu^2(m)}{\phi(m)}\geq \frac{\phi(d)}{d}(\log[x]+1)$

Show that $$\sum_{m\le x , (m,d)=1}\frac{\mu^2(m)}{\phi(m)}\geq \frac{\phi(d)}{d}(\log[x]+1)$$ I want to use the fact that $$\sum_{m\le x , (m,d)=1}\frac{\mu^2(m)}{\phi(m)}=\sum_{m\le x} \sum_{r|(m,d)...
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159 views

Asymptotic formula for Prime Numbers

My question is: Obtain an asymptotic estimate for the sum $$S(x)=\sum_{x<p \leq 2 x} \frac{1}{p}$$ with relative error $1 / \log x$ (i.e., an estimate of the form $$S(x)=f(x)(1+O(1 / \log x))$$ ...
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1answer
37 views

odd prime factors of the form $10k+1$ of a number of the form $20n-4$

Let, $ N= (19 \cdot 29 \cdot 59...\cdot q)^{2} - 5.,$ where the primes numbers of the form $10n+9$ has been considered in the product. Then $N$ is of the form $20n-4$ and has odd prime factors in the ...
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1answer
61 views

Upper bound for prime product under Riemann hypothesis

Theorem 8 of Rosser and Schoenfeld in their paper "Approximate formulas for some functions of prime numbers" says that for x>285, $$\prod \limits_{p \leq x} \frac{p_i}{p_i-1} \leq e^\...
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2answers
34 views

Is $d(n)=n^{o(1)}?$ where d is the divisor function. [closed]

I only know that $d(n)=o(n^{\delta})$, for any $\delta> 0$. Dose it imply $d(n)=n^{o(1)}?$
1
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1answer
29 views

Is every modular function on $\Gamma$ univalent?

Suppose $f$ is a modular function on $\Gamma$ then it has a fundamental region $R_L$. Since $f$ is modular, it can be expressed as a rational function of $J(\tau)=\frac{g_2^3(\tau)}{\Delta(\tau)}$. To ...
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63 views

Quaternionic and octonionic analogues of the Basel problem

It is a well-known fact that $$\sum_{0\neq n\in\mathbb{Z}} \frac{1}{n^k} = r_k (2\pi)^k$$ for any integer $k>1$, where $r_k$ are rational numbers which can be given explicitly in terms of Bernoulli ...
7
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1answer
89 views

Deciding convergence/divergence of $\sum_{m \geq 1} \frac{1}{m^3} \sum\limits_{\substack{k=1\\(m,k) = 1}}^m k \sin\left(\frac{2\pi k n}{m}\right)$

Let $n$ be a positive integer. I am attempting to determine whether the series $$ \sum_{m \geq 1} \frac{1}{m^3} \sum_{\substack{k=1\\(m,k) = 1}}^m k \sin\left(\frac{2\pi k n}{m}\right) $$ converges or ...
3
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1answer
82 views

Behmann's proof of Infinitude of primes.

I am having difficulty in understanding the proof of Behmann of Infinitude of primes. Can someone please explain the last part 'The proof is concluded by noticing....' which is in page $178$? Any help ...
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2answers
41 views

Problem related to inequality for prime counting function

Let $p$ prime and define function $f$ as $$f(n)=\sum_{p-1\mid2n}1$$ Claim Let $\pi(n)$ shows prime counting function Can it be shown that $$\pi(n)\sim f(n)$$ And that there exist real number $a$ such ...
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0answers
33 views

what is $\#\{x\in \mathbb{F}_p: \gcd(x,d)>p^{\epsilon}\}$? for $\epsilon>0$ arbitrary and some d.

Can we say that $\#\{x\in \mathbb{F}_p: \gcd(x,d)>p^{\epsilon}\}\leq p^{o(1)}$?
0
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2answers
88 views

$\sum_{p\leq x} \frac{1}{p}$ ~$ \log \log x$ as $ x\to \infty$.

I have heard that, Euler proved $\sum_{p\leq x} \frac{1}{p}$ ~$ \log \log x$ as $ x\to \infty$. Can anyone please refer me the paper or any link where I can find the proof? I know there are standard ...
1
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0answers
21 views

Existence of pair of primes with difference $2n$ for every integer $n$ [duplicate]

So this might be a silly question. There is the open problem of proving there are infinitely many pairs of primes with a fixed even difference, for any given even number. This maybe implies that it's ...
2
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1answer
81 views

Understanding Wittgenstein's proof of Infinitude of prime

Can someone please tell me why the last claim "It is thus the case..." is true? I tried considering negation of the last claim. But it didn't help. Any help would be appreciated. Thanks in ...
0
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1answer
39 views

Calculating the residue of given quantity

The attached paper claims on top of page 4 that \begin{align} Res(\zeta^2(s)\prod_{p|d}(1-p^{-s})^2\frac{x^{s-1}}{s}; s=1) = \frac{\varphi^2(d)}{d^2}(\log{x}+2\gamma-1)-\frac{2\varphi(d)}{d}\sum_{\...
2
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1answer
80 views

Elementary proof of existence of a prime in an arithmetic sequence

Let $a,n$ be positive integers with $(a,n) = 1$. Then by Dirichlet's theorem on primes in an arithmetic progression we know that there are infinitely many primes $p$ satisfying $p\equiv a\pmod n$. ...
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47 views

Dirichlet series and analytic properties of w(n)

w(n)=v(n) is the number of distinct prime factors of n. i have found Its Dirichlet Series and Abscissa of Convergence from Apostol's book. Proof. Let $a_{n}$ indicate whether $n$ is prime. For $\sigma&...
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1answer
31 views

Concerning Gauss Sums

I am stuck at the following exercise: Given a character $\chi$ modulo $q$, define the Gauss sum $$\tau(\chi) := \sum_{a=1}^q\chi(a)e\bigg(\frac{a}{q}\bigg) $$ where $e(x) := e^{2\pi ix}$. Show that $$...
0
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1answer
28 views

Show that $ \Phi(x,z) = x^{\delta} \cdot \prod_{p \le z} \bigg(1-\frac{1}{p^{\delta}}\bigg)^{-1}$

I am stuck at the following exercise: Let $\Phi(x, z)$ be the number of $n \le x$ all of whose prime factors are less than or equal to $z$. Prove that for any $\delta > 0$ holds $$ \Phi(x,z) \le x^...
5
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1answer
112 views

Do the numbers preceding primes have on an average fewer divisors than the numbers succeeding primes?

I wanted to see if the numbers preceding primes behaved differently in any way form the numbers succeeding primes so I calculated at the average number of divisors of number of the form $p-1$ and $p+1$...
1
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1answer
76 views

Show that $\sum_{n \ge x} \frac{\chi(n)}{\sqrt{n}} = \mathcal{O}\bigg(\frac{1}{\sqrt{x}}\bigg)$

I am stuck at the following exercise: Let $\chi$ be a non-principal character modulo $q$. Show that $$\sum_{n \ge x} \frac{\chi(n)}{\sqrt{n}} = \mathcal{O}\bigg(\frac{1}{\sqrt{x}}\bigg)$$ My Attempt:...
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0answers
39 views

Reference for Euler's Introductio in Analysin Infinitorum

In the following answer it has been claimed that " The reference here is not to Euler's 1737 "factorization" of the harmonic series but to 1748 Introductio in Analysin Infinitorum, ...
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0answers
62 views

How many distinct prime factors are there in the numbers between two primes?

Let $p$ and $q$ be two consecutive primes and $f(p)$ be the number of distinct prime factors of the product $(p+1)(p+2)\cdots (q-1)$. Thus $f(p)$ is a count of the number of distinct primes factors ...
2
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0answers
103 views

There are more primes than square

Will someone please help me in understanding the claims of corollary $2$ and $3$? I understand that corollary $2$ means for large $N$ there are more primes than squares in the interval $[1,N]$ which ...
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0answers
44 views

Where can I find the following papers [duplicate]

Can someone help me by providing the link of the following papers where $2,3,4$ are written by Euler and $5$ th one is by Kronecker.I could not find them in web. Any help would be appreciated. Thanks ...
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1answer
70 views

Where to find English translations of Euler's collected works?

I will highly appreciate if someone can provide me the link to the book or link to the paper where I can find the following papers $(90-94)$ with English translation. Any help would be appreciated. ...
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0answers
21 views

Non-asymptotic bounds for prime counting function for arithmetic progressions

Are there non-asymptotic bounds for prime counting function $\pi_{a,b}(x)$ for arithmetic progressions $a+kb$, where $a$ and $b$ are coprime? I found asymptotic bounds for $\pi_{a,b}(x)$ and non-...
2
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1answer
92 views

Complex Numbers : Why stop at 2 dimensions?

Why is it enough to have extended the numbers to include only one orthogonal imaginary axis? I am wondering in the context of roots of polynomials. I know that the orthogonality of imaginary axis w.r....
6
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0answers
103 views

Why does the infinitude of primes of a certain form matter? [closed]

A result like Dirichlet's theorem makes sense to me as worthy of attention because arithmetic progressions are a natural mathematical phenomenon. But then there are results like the Friedlander-...
0
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1answer
67 views

$ \pi (N)< B \ \frac{N}{\log N }$ where $ B\leq 4.$

Let $p_{1}, p_{2},..., p_{r}$ are all the pimes not exceeding $ N$. We can say $ r< B \frac{N}{\log N }$ where $ B\leq 4$ . I know Chebyshev proved the stronger version of it with $ B= 1.1055...$ ...
3
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2answers
73 views

To check whether $ 10^{c} \frac{N}{e} > (N!)^{\frac{1}{N}} > \frac{N}{e}$

Is the following true or false for $N\geq 3$? $\log N - \log e + c >\frac{\log N!}{N}> \log N - \log{e}$ for some $ c< \log e$ where the logarithm is w.r.t. base $10.$ Calculating some values ...
3
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0answers
46 views

Something related to Dedekind zeta function

Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers, $\mathcal{O}_K^{\times}$ its group of units, $\mathcal{Cl}(K)=J_K/P_K$ its ideal class group. As you know, the Dedekind zeta function ...
-1
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1answer
67 views

An upper bound whose value is known for the series $ \displaystyle{\sum_{p \text{ prime}}}\frac{1}{p^{3/2}}$

We know, series $ \displaystyle{\sum_{p \text{ prime}}}\frac{1}{p^{3/2}}$ is Convergent since it is bounded above by the Convergent series $ \displaystyle{\sum_{n=1}^{\infty}}\frac{1}{n^{3/2}}$. But ...
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0answers
42 views

$ \sum_{p } \sum_{k=2}^{\infty} \frac{\log p}{p^k}$ is Convergent

How can I show the sum $ \sum_{p } \sum_{k=2}^{\infty} \frac{\log p}{p^k}$ where $p$ varies over all the primes, is Convergent? I tried comparing with the convergent series $\sum_{p} \frac{1}{p^2}$, ...
-1
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1answer
34 views

approximating a geometric sum of exponentials by an integral

Let $e(x) = e^{2 \pi i x}$. Let $I = [a,b]$ be a closed interval of real numbers. I am interested in the sum $$ S = \sum_{n \in I \cap \mathbb{Z}} e(\alpha n) $$ for some real number $\alpha.$ Do ...
0
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0answers
29 views

Estimates for this divisor sum

I am reading a paper in analytic number theory. I'm stuck on a line. The author did this which I don't understand. $$\sum_{\substack{q | P \\ v(q) = \gamma}} |\mu(q)|(\frac{x}{hq} + 1) \ll \frac{x}{h}\...
3
votes
1answer
119 views

Missing flaw in finding all integers satisfying $\varphi(n)=n/2$, where $\varphi$ is the Euler totient function

I was reading this book on analytic number theory by Tom M. Apostol, and I came across this problem that asks for all integers that satisfy the following equality: $$ \varphi(n) = n/2$$ where $\...
0
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0answers
23 views

What is good reference for circle method (Analytical number theory)??

I am trying to read the circle method from 'A Course in Analytic Number Theory by Marius Overholt.'$\ $ This seems a little hard. any other reference would be greatly appreciated.
2
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0answers
42 views

Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n}$

Dirichlet series are an important area of research in Analytic Number Theory, and their values at $s=1$ (or in general on the edge of their abscissa of convergence) are generally of special importance....

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