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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2
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0answers
52 views

Understanding the Hurwitz-Kronecker Class Number Formula

The Hurwitz-Kronecker Class Number is given by the formula $H(d)=\sum_{Q\in Q_d/(\Gamma=PSL_2(\mathbb{Z}))}\frac{1}{w_Q}$ where $w_Q=card(stab(\alpha_Q))$ with $\alpha_Q$ being the unique zero ...
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0answers
30 views

Don't know what to do with little-oh!

Let $f:\mathbb{N} \to \mathbb{C}$ be a function for which there exists a positive constant $A$ such that \begin{equation} \lim_{x\to \infty} \frac{1}{x}\sum_{n\leq x} f(n) = A. \end{equation} Prove ...
2
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2answers
163 views

A fun fact relating to Goldbach Conjecture

I have noticed a fact when verifying the Goldbach Conjecture. Let $n$ be an even number larger than 6, we can easily write $n=i+j$, where $i$ and $j$ are both prime numbers. Now let $i\le j$, and ...
2
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1answer
152 views

A combinatorial sum and identity involving Stirling numbers of the second kind

Let $n, k \geq 1$. Let $a(j),\, 1\leq j \leq k$, be a sequence of real numbers. Consider the sum $$ \sum_{j=1}^k j! S(k, j) {n \choose j} a(j), $$ where $S(k,j)$ are Stirling numbers of the second ...
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1answer
28 views

Proof Verification: Let $f(\chi)$ be the conductor of $\chi$, proof that $f(\chi)=f(\chi_1)\cdots f(\chi_r)$.

This is a detailed problem, let me write down the problem and the process I have done: Assume that $k=k_1k_2\cdots k_r$ where $k_i$ and $k_j$ are relatively prime for $i\neq j$. Let $\chi$ be a ...
2
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1answer
44 views

If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.

I'm reading the proof of the multiplicative property of $$s_k(n)=\sum_{d|(n,k)}f(d)g\bigg( \frac kd\bigg)$$ The book wrote that in order to understand the proof, we need to know if $a,b,k,m$ are ...
6
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1answer
176 views

Density of integers $n$ with all prime factors of order $O(\log n)$?

For a rational integer $n \in \mathbb{Z}_{+}$, let $\mathfrak{p}(n)$ denote the set of (distinct) prime factors of $n$. Then for a positive constant $c$, let $$f(x) = \vert\{n\in\mathbb{Z}_{+}:\ n\...
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1answer
58 views

Approximation of $\zeta$(s) at $s=1$

I am currently taking an Analytic Number Theory unit and we're working on the zeros of the zeta function. In the proof of $\zeta(1+\textit{i}t) \neq 0$ for $t \in \mathbb{R}$, we suppose that $\zeta$(...
2
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0answers
34 views

Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms

Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients. (1) Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know ...
3
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1answer
131 views

Statement Equivalent to the Riemann Hypothesis

I am told that the Riemann Hypothesis is equivalent to the condition: $\psi(x) = x + O(x^{1+o(1)})$, and asked to prove this in the forward direction. (Here $\psi(x)$ is the Chebyshev Function). ...
1
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1answer
753 views

Rational vs irrational [closed]

If two points on a number line is shown, are rational numbers between the two points is more or irrational number is more ? I have tried using probability , my collegue who was like my teacher also ...
1
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1answer
399 views

Proving the Bernoulli number relation $(1+B)^n=B^n$

We know that we can generate the Bernoulli numbers using the relation $(1+B)^n=B^{[n]}$ where $B_n$ is $n$th Bernoulli number. But how we can prove this works? Thanks to all. Edit 2: is there a ...
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0answers
51 views

Estimating $\sum\limits_{n\leq x} d_3(n)$.

If $d_3(n)$ denotes the number of ways to write $n$ as a product of $3$ positive integers then how do I show that as $x\to \infty$, $\sum\limits_{n\leq x}d_3(n)=\frac{x(\log x)^2}{2}+O(x\log x)$. ...
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0answers
221 views

An Engineer sets out to Prove Fermat's Last Theorem …

This started off as a joke post of mine on a Facebook Group called "Bad Maths that Gives the Right Answer", in which I pulled a Fermat and claimed that the last bit of the proof was too long to post. ...
1
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1answer
30 views

A not very obvious question about $\{h+tk\}$ sequence.

Let $h$ and $k$ be positive integers such that $\gcd(h,k)=1$. Let $A(h,k)$ be the sequence $$A(h,k)=\{h+kx|x=0,1,2,3,\cdots\}.$$ Let $S$ be a infinite subset of $A(h,k)$, prove that for each positive ...
1
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1answer
76 views

The limit on the ratio of the Dirichlet eta functions

If we accept-- $s_o$ -- as one of the non-trivial zeros of the Riemann zeta function by $0 <Re(s_o)<1$ and $Re(s_o)$ is the real part of a complex variable, we know: $$\eta(s_o ) = \...
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0answers
36 views

Does the first Hardy-Littlewood conjecture imply $\sum\limits_{r=1}^{n-3}\Lambda(n-r)\Lambda(n+r)\sim 2C_{2}n$?

Hardy-Littlewood conjecture predicts that the number of Goldbach decompositions $p+q=2n$ should be asymptotically equal to $K\frac{n}{\log^2 n}\prod\limits_{p>2,p\mid n}\frac{p-1}{p-2}$ for a ...
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0answers
34 views

Show $\frac{\sum_{n=1}^{\infty}\frac{1}{\phi(n)^s}}{\zeta(s)}=\prod_{p}(1-1/p^s+1/(p-1)^s)$.

I want to show that the following equality and that the product is absolutely convergent and uniformally convergent on compact subsets of ${s:Re(s)>1}$. $$\frac{\sum_{n=1}^{\infty}\frac{1}{\phi(n)^...
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1answer
26 views

Solutions for $n$? Use Stirling approximations if needed

$$(2n)! = a^{2n}$$ where $a \in \mathbb R$, and $n \in \mathbb N$. This is relevant because of a research question I'd asked and received an answer to by Sotiris here
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5answers
3k views

How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?

I mean the Laurent series at $s=1$. I want to do it by proving $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$, based on the integral formula given in Wikipedia. ...
1
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1answer
50 views

Number of integers coprime to a given integer $q$ in some range $[x, x+y]$

I am asked to show that for $1 \leq x,y$ and an integer $q$, we have: $S(x,x+y,q) = |\{x < n \leq x + y \mid n \text{ is comprime to } q\}| = \frac{\phi(q)y}{q} + O(2^{\omega(q)})$, where: $\...
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1answer
37 views

About absolute convergence of complex series

I know I need to incorporate uniqueness theorem of Dirichlet series to get some kind of contradiction, but don't know how to proceed?
2
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0answers
64 views

Convergence of $\sum\limits_p \frac{\chi(p)}{p}$ and the prime number theorem

Consider the sum $$\sum\limits_p (-1)^{\frac{p-1}{2}}\frac{1}{p}=-\frac{1}{3} + \frac{1}{5} - \frac{1}{7} - \frac{1}{11} + \frac{1}{13} + \frac{1}{17} - \cdots \tag{1}$$ of signed reciprocals of the ...
9
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0answers
725 views

How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$

I'd like to simplify $$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form $$...
2
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1answer
91 views

Prove that $\sum\limits_{n=1}^{N} \frac{1}{n} \le \text{exp}\left(2\sum\limits_{n=1}^{N} p_k^{-1}\right)$

I want to proof that $$\sum_{n=1}^{N} \frac{1}{n} \le \text{exp}(2\sum_{n=1}^{N} p_k^{-1})$$ where the sequence $(p_n)_n$ denotes the prime sequence. I tried to do this by induction over N, ...
2
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0answers
31 views

The inverse of a Dirichlet product is the Dirichlet product of the inverses of each function

Let $f,g$ be arithmetic functions. According to Wikipedia, $(f*g)^{-1} = f^{-1} * g^{-1}$ if $f(1) \neq 0$ and $g(1) \neq 0$. However, it is not clear to me why this is true, and the statement does ...
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0answers
65 views

Lower bound for number of primes up to $x$.

It is possible to prove the statement that $\pi(x)$, the number of primes up to $x$ is bigger than $\sqrt x$ for $x \ge 3$, in an elementary way? More generally can we prove that for every $0<\...
3
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2answers
211 views

$\sum_{n \leq x} \frac {\log n}{n^3} = C - \frac {\log x}{2x^2} -\frac {1}{4x^2} +O \bigg (\frac {\log x}{x^3} \bigg)$

I try to prove the following equation: for $x \geq 2$ $$\sum_{n \leq x} \frac {\log n}{n^3} = C - \frac {\log x}{2x^2} -\frac {1}{4x^2} +O \bigg (\frac {\log x}{x^3} \bigg)$$ To my opinion I have to ...
0
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1answer
79 views

Show that $x \left(\sum\limits_{n\leq x} \frac{\ln(n)}{n^2}\right) +\theta(x)=O(x)$

I know that $$x\left(\sum\limits_{n\leq x} \frac{\ln(n)}{n^2}\right)+\theta(x) \leq x\left(\sum\limits_{n\leq x} \frac{\ln(n)}{n}\right)+\theta(x)=x\frac{1}{2}\left(\ln{x}\right)^2+Cx+O(\ln x)+O(x)$$ ...
0
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0answers
17 views

Fundamental approximation in the circle method

I am interested in the circle method and I am currently working on Vaughan's book. Let $f$ be the generating function $f$ of the squares, that is to say the power series sum of $z^{m^2}$. One of the ...
1
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1answer
43 views

Hypergeometric functions and modular forms

May I please ask if it is possible to write Hypergeometric functions in terms of Jacobi theta functions? I am trying to bring the following Hypergeometric expression (pg.9, eq 4.3) into a known ...
1
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1answer
49 views

Generalized Jacobi theta functions - Laurent series expansion of H(w,q,S).

Can someone please assist me with the missing steps in the proof of 'proposition 3' in M. Kaneko and D. Zagier paper (https://people.mpim-bonn.mpg.de/zagier/files/progmath/129/165/fulltext.pdf, pg. 4)....
2
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1answer
36 views

Bounding a Mobius Fractional Sum

How does one show the following estimate: $\displaystyle \bigg\lvert 1+\sum_{n \leq x} \mu(n) \Big\{\frac{x}{n}\Big\} \bigg\rvert \leq x$, where $x\geq1$ ? My attempt was to use the triangle ...
0
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0answers
45 views

reference request - moments zeta function/dirichlet polynomials

I would like to study some material on the moments of the Riemann Zeta function and Dirichlet polynomials (mean value theorems). I was looking both for some introductory material and for some more ...
3
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1answer
64 views

What is the probability that a large integer has at least one small factor? [closed]

I have large integer $N$. What is the probability that it has at least one factor that is less than $B$?
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0answers
51 views

Estimate about $\sum_{n\leq x}\mu(n)\log \frac xn$

I'm finding the estimate of $$\sum_{n\leq x}\mu(n)\log \frac xn$$ There is a formula saying that $$\sum_{n\leq x}f(n)G\bigg(\frac xn\bigg)=\sum_{n\leq x}g(n)F\bigg(\frac xn\bigg)$$ where $F(x)=\sum_{n\...
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0answers
50 views

Prove the equivalent form of the Selberg's Formula $\psi(x)\log x+\sum_{p\leq x}\psi\bigg(\dfrac xp\bigg)\log p=2x\log x+\mathcal O(x).$

This is the Selberg's Formula: $$\psi(x)\log x+\sum_{n\leq x}\Lambda(n)\psi\bigg(\dfrac xn\bigg)=2x\log x+\mathcal O(x).$$ In the book Apostol Analytic Number Theory, the exercise tells me to use this ...
7
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1answer
487 views

Modular transformations of $\eta(\tau)$

Under a modular transformation the Dedekind $\eta$ function transforms as $$\eta(-1/\tau) = \sqrt{-i \tau}\eta(\tau) \, .\tag*{$(*)$}$$Siegel gives a proof in this paper here that uses complex ...
0
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0answers
31 views

Portion of prime congruent to $a$ mod $m$ in the factorial of $n$

I'm learning analytic number theory and find this interesting question from one textbook: It is well known in analytic number theory that: \begin{equation} \sum_{\substack{p\leq x \\ p\equiv a \...
1
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1answer
39 views

How to prove that $\text{End}_{\mathbb{F}_p}(E)$ is commutative for a given elliptic curve E?

Given a prime $p$ and considering the finite field $\mathbb{F}_p$, I need to see that $\text{End}_{\mathbb{F}_p}$(E) is commutative using orders. It is known that $\text{End}_{\mathbb{F}_p} \subseteq \...
1
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0answers
47 views

Square roots of the unity (DIrichlet convolution)

I am having a little trouble with this question. Given an arithmetic function f, a “Dirichlet square root” of f is an arithmetic function g such that $g ∗ g = f$. Prove by elementary techniques that ...
1
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2answers
28 views

Let $a_1< \cdots< a_n\leq x$, where no $a_i$ divides product of others, show that $n\leq \pi(x)$.

Let $a_1< a_2< \cdots< a_n\leq x$ be a set of positive integers such that no $a_i$ divides the product of the others. Prove that $n\leq \pi(x)$. I have tried to argue by contradiction, ...
10
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3answers
354 views

M and n are positive integers such that $2^n - 3^m > 0$. Prove (or disprove) that $2^n - 3^m \geqslant 2^{n-m}-1$.

Given that $2^n - 3^m > 0$, I know that $n > m\log_{2}3$ (*). If $2^n - 3^m \geqslant 2^{n-m}-1$, $n>= m + \log_{2}\frac{3^m-1}{2^m-1}$ (**). This is the result when I graph it out ($m$ -> $...
1
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0answers
18 views

Existence of Dirichlet Inverse.

How do we show the existence of the Dirichlet Inverse of a function $f$? I know using induction that we can find a recursive form of $f^{-1}$. But I can't seem to show the existence. Any ideas? Most ...
0
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0answers
23 views

Calculate the limit of special sequence.

There is given a sequence $A_{0},A_{1},...$ wich are all ordered zeros of Mertens fuction $M(n)$. How to find the following limit ?: $$\lim_{n\to\infty}\frac{A_{n+1}-A_{n}}{A_{n}^{0.75}}$$ Does ...
0
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0answers
47 views

Why is it impossible to invert the analytic continuation of a Dirichlet series?

By Mathematica (and the truncated Euler MacLaurin formula) I know that: $$\zeta(s)=\lim_{k\to \infty } \, \left(\sum _{n=1}^k \frac{1}{n^s}+\frac{1}{(s-1) k^{s-1}}\right) \tag{1}$$ when the real part ...
1
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0answers
32 views

What is sum of squared elements of Farey sequence?

Let's consider a Farey sequence: $$F_{n}=\{a_{1},...,a_{k}\}$$; Where given elements satisfy definition of $n$-th Farey sequence. My problem: Find the formula for the following sum: $$\sum_{l=1}^{k}...
3
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1answer
132 views

Dirichlet convolution of multiplicative functions

Are there two nonzero arithmetic functions,say $f,g$, which are not multiplicative but their Dirichlet convolution is multiplicative?
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1answer
43 views

Inverse of completely multiplicative function with respect to dirichlet convolution

Is the inverse of a completely multiplicative function $f(n)$ with respect to Dirichlet convolution again completely multiplicative? I know that for multiplicative functions its true(Apostol's ...
-1
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1answer
61 views

Big Oh of values of Riemann zeta function

There is a equality in a proof in Apostol's Analytic Number Theory as follows: $O(x^{\alpha} \zeta(\alpha)) = O(x^{\alpha})$ for arbitrary real number $\alpha \ge 0$. How do we say that? Does ...