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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8
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1answer
224 views

Relationship between GCD, LCM and the Riemann Zeta function

Let $\zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased, $$ \frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\text{lcm}(k,i)}\bigg)^s \approx \...
0
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0answers
33 views

Show, $\sum_{\chi\neq\chi_0}(\sum_{a=2}^{p-2}\sum_{b=1}^{p-1}\chi(a)(\frac{b^2-a^2}{p})(\frac{b^2-1}{p}))^2|L(1,\chi)|=O(p^{7/2}\log p)$.

I can show, it is bounded by $p^4\log p$. But I think the bound should be smaller.
0
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0answers
36 views

Can a upper bound of $\sum_{b=1}^{p-1}\left(\frac{b^2-a^2}{p}\right)\left(\frac{b^2-1}{p}\right)$, $a\in{Z}$, be strictly less than $\sqrt{p}$?

By weil estimate I can only say, one bound is $\sqrt{p}$. Can it be strictly less than $\sqrt{p}$? I want to see whether one better bound be given or not.
1
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0answers
42 views

Why is the series representation of the logarithm of the zeta function analytic?

I try to prove, that the logarithm of the $\zeta$-function has the following representation for $z \in \mathbb{C}$ with $\text{Re}(z) > 1$ $$\log\zeta(z)=-\sum_p\log\left(1-\frac{1}{p^z}\right).$$ ...
0
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0answers
24 views

Probability for an L-function to be RS-primitive

Assuming an L-function is any element of the intersection $\mathcal{L}$ of the Selberg class $\mathcal{S}$ and the class of automorphic L-functions $\mathcal{A}$, define the notion of Galois class of ...
0
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1answer
72 views

Is $\sum_{n=1}^{\infty} \frac{\mu(n)}{n} = 0$? [duplicate]

Does $\sum_{n=1}^{\infty} \frac{\mu(n)}{n}$ converge and does it converge to $0$ ?. I know that $\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}$ converges on $Re(s) > 1$ to $\prod_{j=1}^{\infty}{(1-\frac{1}...
0
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1answer
60 views

Minimum and maximum of a partial Euler product?

Question: If if $n\in\mathbb{N}$ and $s\in \mathbb{C},$ say $s=\sigma+t\sqrt{-1},$ then Dirichlet Beta function is defined to be $$ \beta(s)=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}; $$ which for ...
0
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0answers
14 views

On Property of Ramanujan Tau function

I have been self studying number theory from Apostol Dirichlet series and modular forms and I could not solve this question from chapter 4 Pg92. There is a solution of this question on stack exchange ...
0
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0answers
20 views

On proving a result on Jacobi theta function

I have been self studying Apostol Dirichlet series and Modular forms and I have could not solve this problem from Chapter 4 - Prove that $\theta(-1/t) = \sqrt{-it}\ \theta(t)$ , t belongs to upper ...
3
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0answers
47 views

Density of Numbers with Exactly One Prime Factor of Multiplicity 1

Let $S$ be the set of positive integers $n$ with the property that exactly one prime factor of $n$ has multiplicity $1$ and every other prime factor has multiplicity greater than $1$ (to be clear, $S$ ...
2
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1answer
59 views

Does the average reciprocal of the smallest or largest prime factors of integers converge?

I have observed the following experimentally. Can this be proved for disproved? Claim: Let $a_n$ and $b_n$ be the smallest and the largest prime factor of $n$ respectively. Then $\dfrac{n}{\sum_{k \...
0
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0answers
29 views

Miswritten exponential sum definition. What did my professor likely mean to write?

In my analytic number theory course, in a lesson about exponential sums, my professor defined a particular exponential sum. I believe that I copied it down correctly, because I scrutinized it for a ...
1
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0answers
69 views

Weyl's equidistribution theorem in the case of rational numbers

Let θ be a non-zero real number and let N be a positive integer. Consider the fractional parts of nθ for 0 ≤ n < N. This creates a maximum of N possible intervals. Show that there as only three ...
0
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1answer
42 views

$p$-adic integers and valuation

Let $p$ be a prime, $x \in \mathbb{Z}_p$. I say that if $x \in \mathbb{Z}_p$ then $\text{val}_p(x) \ge 0$ as follows: Let $x = a_0 + a_1 p + a_2 p^2 + \dots$. Then, $\text{val}_p(x) \ge \text{min}(\...
1
vote
1answer
102 views

Why do the coefficients of this series grow like a polynomial?

Set-up Let $f: \mathbb{N} \to \mathbb{C}$ be a multiplicative arithmetic function (meaning that $f(mn) = f(m)f(n)$ whenever $m$ and $n$ are relatively prime). Suppose that for every $\epsilon >0,...
1
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1answer
35 views

Ring of p-adic integers and harmonic sums

I am trying to understand p-adic integers $\mathbb{Z}_p$ and want to ask two questions: For a fixed rational number $\frac{a}{b}$ and a prime $p$, I can write $p$-adic expansion of the fraction after ...
0
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1answer
57 views

Are these equalities wrong $\sum_{n\le x}\mu(n)o(\frac xn)=o(x\sum_{n\le x}\frac 1n)=o (x\log x) $?

I simply found an asymptotic relation for $\sum_{n\le x}\mu(n)o(\frac xn)$ like below: $$\sum_{n\le x}\mu(n)o(\frac xn)=o(x\sum_{n\le x}\frac 1n)=o (x\log x) $$ But Basil Gordon, an American ...
1
vote
1answer
57 views

Interlacing with arithmetic progressions?

Given integers $0<a_1<\dots<a_t$ and $0<b_1<\dots<b_t$ with $a_t<b_t<M$ can we find integers $m,n,m',n'\in\mathbb Z$ such that $$ma_i+n<m'b_i+n'<ma_{i+1}+n$$ holds at ...
3
votes
3answers
84 views

Is there a positive integer $n \ge 2$ for which $\frac{k}{\pi(k)} = n$ has no solution?

For a given positive integer $n \ge 2$ let $a_n$ be the number of integers $k$ such that $\dfrac{k}{\pi(k)} = n$ where $\pi(x)$ is the prime counting function. The first few values of $(n,a_n)$ are $$...
1
vote
1answer
28 views

Chebyshev functions limsup

Let $\delta=\limsup\frac {\psi (x)}x $ i want to prove that $$\sum_{n\le x}\psi (\frac xn)\le (\delta+\epsilon)x\log x+x\psi (N) $$ Given $\epsilon>0$ there is $N$ such that $x>N$ implies that ...
0
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0answers
34 views

Davenport Multiplicative Number Theory, pp. 19

Recently I have been reading the book in the title of this question to brush my interest in math up again in the course of getting a certificate for teaching math in middle schools. I know some ...
1
vote
0answers
24 views

Imprimitive Dirichlet characters as sum of additive characters

Let $\chi$ be a Dirichlet character modulo $q$. Let $e(x)=\text{exp}(2\pi i x)$ and $\tau(\chi)$ be the Gauss sum $\tau(\chi)=\sum_{m \,\text{mod}\, q}\chi(m)e(m/q)$. For any $n$ with $\gcd(n,q)=1$, ...
0
votes
2answers
58 views

An inequality involving Euler’s $\varphi$-function

Let $\varphi$ be Euler’s phi-function. I have seen it claimed that $$\varphi(n)/n = O(\log \log n).$$ Could someone either give a proof of this fact or tell me a reference where I can find this proof?...
3
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0answers
409 views

A Special Observation on Prime Numbers and $\pi (n)$

$\eth(n)$ is a little algorithm I made, which may appear to be quite complex, so I will start with an example middle of the post. Questions are at the end of the post. Definition Let $...
1
vote
1answer
51 views

Unique Structure in base for power $1,2$and$3$

Let's $1<a\in\mathbb{N}$ And $$A^{k}=\sum_{i=1}^{a}i^{k}$$ Here $t $ is a number from any base $q$ can be converted in base $b$ written as $$(t)_{q}=(b_{r} b_{r-1} ... b_{2} b_{1})_{b}$$ Now ...
1
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0answers
34 views

Interchange the order of summation in Dirichlet convolution

I'm doing the following exercise from Apostol's number theory book, a solution, or better a Hint would be welcome. (Chapter 2 ex 14) Let $f(x)$ be defined for all rationals $x$ in $0 \leq x \leq 1$ ...
0
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1answer
33 views

What is $O(o(x))$ as $x\to\infty $? I think it is $o (x)$ or $O (x^\delta) $ where $\delta <1$

Let $M (x)=\sum_{n\le x}\mu (n) $. We know that $$M (x)\log x+\sum_{n\le x}M (\frac xn)\Lambda (n)=O (x) $$ I want to prove $$M (x)\log x+\sum_{p\le x}M (\frac xp)\log p=O (x) $$ We have $$M (x)\...
-1
votes
0answers
14 views

Limited Concentration of Singularities at Roots of Unity of a Power Series with Bounded Coefficients

Let $V$ be a set of positive integers, and let: $$\varsigma_{V}\left(x\right)\overset{\textrm{def}}{=}\sum_{v\in V}x^{v}$$ Let $T_{V}$ denote the set of all $t\in\left[0,1\right)\cap\mathbb{Q}$ such ...
-1
votes
1answer
15 views

A Converse For A Particular Case of the Hardy-Littlewood Tauberian Theorem

Let $V$ be a set of positive integers, and let: $$\varsigma_{V}\left(x\right)\overset{\textrm{def}}{=}\sum_{v\in V}x^{v}$$ Defining the natural density of $V$ by the limit: $$d\left(V\right)\overset{\...
0
votes
3answers
84 views

Why doesn't the Riemann Zeta Function have a zero at $s=0$?

I have a few questions regarding the Riemann Zeta Function and its zeros. First, I will state what is clear to me. I see that $\zeta (s)$ is clearly defined and nonzero when $Re(s)>1$ since then $\...
2
votes
1answer
50 views

generating function of sum of divisors function

It is well known that the function $$\sigma_k(n)=\sum_{d|n}d^k$$ has a generating function. For a number field $K$, suppose that $\mathfrak{a}, \mathfrak{b}$ are ideals in some ideal class $C$ and ...
1
vote
0answers
35 views

asymptotic notation estimate

I'm trying to arrive at estimate 1.17 (page 21) of Koukoulopoulos lecture notes [https://dms.umontreal.ca/~koukoulo/documents/notes/sievemethods.pdf] $$\#\{n \leq x : p|n \Rightarrow p > \sqrt{x}\}...
0
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1answer
26 views

Show that $\int_1^{\infty}e^{-\pi x}(x^{\sigma/2} + x^{(1-\sigma)/2}) x^{-1} dx \ll (1 + |\sigma|)^{\pi |\sigma|} $

We must show that $$\int_1^{\infty}e^{-\pi x}(x^{\sigma/2} + x^{(1-\sigma)/2}) x^{-1} dx \ll (1 + |\sigma|)^{\pi |\sigma|}. $$ Here is my attempt, however I wondered if there is a one-trick wonder ...
1
vote
1answer
40 views

Gamma Function Integral Identity

On p.11 of D. Bump's "Automorphic Forms and Representations" he uses the following identity in a proof of the functional equation of a Dirichlet $L$-function: $$ \int_0^\infty e^{-\pi tn^2}t^{(s+\...
2
votes
1answer
143 views

sum over reciprocal of primes times coefficient

I would like to show that $$ \sum_{p\leq x} \frac{1}{p^{1+2/\log x}}\left(\frac{\log\left(x/p\right)}{\log(x)}\right)^2=\log\log x +\mathcal{O}(1) $$ What I have tried Since we know that $$ \sum_{p\...
1
vote
1answer
28 views

Define a multiplicative function $r(n)$ by $r(p^a)=\frac{\binom{2a}{a}}{4^a}$. Show that $(\sum_{n\leq p^{3/2}}\frac{r(n)}{n})^2<< \log p$

Define a multiplicative function $r(n)$ by $r(p^a)=\frac{\binom{2a}{a}}{4^a}$. Show that $$\left(\sum_{n\leq p^{3/2}}\frac{r(n)}{n}\right)^2<< \log p$$ I am confused about, how $\log(p)$ is ...
0
votes
1answer
28 views

Elementary number theory in modular functions

I am self studying Apostol Dirichlet series and Modular functions in number theory and I am struck on Ch -2 Problem 17. It is a problem of elementary number theory but I am not able to think about it. ...
0
votes
0answers
30 views

Logarithmic derivative of Dirichlet series and relation to Mangoldt function

On Wikipedia, the following formula is given: $$\frac {F^\prime(s)}{F(s)} = - \sum_{n=1}^\infty \frac{f(n)\Lambda(n)}{n^s}$$ I understand the proof for the zeta-function, but not for the Dirichlet ...
-1
votes
2answers
40 views

Generalised sum of divisors.

Let $$F(x,k)=\sum_{a_{1}a_{2}...a_{k}\le x}1$$ Find asymptotic formula for $F(x,k)$ For example it is known. $$F(x,2)=\sum_{n\le x}\tau(x)=x\log x+(2\gamma-1)x+O(x^{\frac{1}{2}})$$ The proof of ...
2
votes
1answer
34 views

Integral countour with Mangoldt function

In an analytic proof for prime number theorem I found a passage I could not understand: $$A=\sum_{n\leq x} \frac{1}{2 \pi i} \int_{c-\infty i}^{c+\infty i} \frac{\Lambda(n) (x/n)^s ds}{s (s+1)} = ...
1
vote
1answer
26 views

Klein function, modular functions

I have been self studying Apostol's Dirichlet Series and Modular Forms and I am stuck on Theorem 2.8 on page 40. It says it is clear that every rational function of J is a modular function. I don'...
0
votes
0answers
49 views

Simpler version of Tchebotarev's theorem (base field $\mathbb{Q}$)

All texts I could find about Tchebotarev theorem prove it for a general Galois extension $L|K$. I wonder if there is a specific proof for the simpler case of $K=\mathbb{Q}$, i.e.: For a given ...
2
votes
2answers
50 views

Regarding reference book in multiplicative number theory

I am self studying analytic number theory and I have finished Apostol's Introduction to Analytic Number Theory just now. Now I have started the second volume of Apostol's Dirichlet Series and Modular ...
1
vote
1answer
118 views

How to prove that$\sum_{n\le x}(\psi(\frac xn)-\vartheta(\frac xn))\Lambda (n)=O (x) $

I'm trying to prove that $$\sum_{n\le x}(\psi(\frac xn)-\vartheta(\frac xn))\Lambda (n)=O (x) $$ After some calculations, l arrived to $$O (\sqrt{x}\log x\sum_{m=1}^\infty x^{\frac 1m} (\frac{1}{\...
3
votes
3answers
171 views

Proof $\sum\limits_{n \le k/2} \frac 1 n < \log k$ to show Pólya's inequality

In Introduction to Analytic Number Theory, theorem 8.21, Apostol's proves Pólya's inequality. The last step of the proof requires showing: $\sum\limits_{n \le k/2} \frac 1 n < log \; k$ my proof ...
0
votes
1answer
51 views

Asymptotic formula for log summation

Let $\varphi(x) := \sum_{n\leq x} \log(n) $. Show that \begin{align} \varphi(x) = x \cdot \log(x) + O(x) \quad \forall\,\, x > 1 \end{align} I think I must show both inequalities, and if it’s the ...
0
votes
1answer
56 views

Formula for the reciprocal of the Hurwitz zeta function

The following equation is for the reciprocal of the zeta function at $Re(s) > 1$: $$ \frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}} = s \int_{1}^{\infty} \frac{M(x)}{x^{1+s}}dx, \ ...
2
votes
1answer
31 views

An Inequality Involving Subsets of $\mathbb{N}$ of Positive Density

Given a set $V$ of non-negative integers, let: $$V\left(t\right)\overset{\textrm{def}}{=}\left\{ v\in V:0\leq v\leq t\right\} ,\textrm{ }\forall t\in\mathbb{R}$$ and let $\left|V\left(t\right)\right|$ ...
4
votes
0answers
115 views

The probability that $s$ integers selected according to the ideal soliton distribution are relatively $r$-prime

Fix integers $r,s\geq1$, not both 1. Definition. We say that integers $a_{1},\ldots,a_{s}$ are relatively $r$-prime if their greatest common divisor has no perfect $r$th power factors > 1. (When $r=...