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

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1answer
52 views

Wintner's mean value theorem

This is an exercise (exercise 2.22 p80) from A.J. Hildebrand's Introduction to analytic number theory (an online lecture notes). Let $g$ be an arithmetic function, and let $f=1*g$ (i.e.,$f(n)=\sum_{d\...
2
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1answer
44 views

Asymptotic estimate for the sum $\sum_{n\leq x} 2^{\omega(n)}$

How to find an estimate for the sum $\sum_{n\leq x} 2^{\omega(n)}$, where $\omega(n)$ is the number of distinct prime factors of $n$. Since $2^{\omega(n)}$ is multiplicative, computing its value at ...
3
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2answers
89 views

Prove that the value of the constant $C$ must be $1$

After proving the prime number theorem in class, our professor directs us to a remark by Lagrange that for large values of $x$, $\pi(x)$ is approximately equal to $$ \frac{x}{\log x - B}. $$ (This is ...
5
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2answers
89 views

Proving that $\pi(2x) < 2 \pi(x) $

In our analytic number theory class we were given the following problem as homework: prove rigorously that for large $x$ the number of primes in $(1,x]$ exceeds that in $(x,2x]$. In class we proved ...
1
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1answer
47 views

Asymptotic estimate of the sum $\sum_{n\leq x}1/\phi^2(n)$

How to show that we have the following estimate: $$\sum_{n\leq x}\frac{1}{\phi^2(n)}=c+O(\frac{1}{x}),$$ where $\phi$ is the Euler's totient function and $c$ is a constant. I tried to use the ...
2
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1answer
39 views

Number of subsets $S$ of $[n]$ such that $\gcd(S)$ is coprime to $m$

Fix positive integers $m,n$. Is there a way to count the number of non-empty subsets $S$ of $[n] = \{1, \ldots, n\}$ such that $\gcd(S)$ is coprime to $m$? Can we come up with an expression for such a ...
1
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1answer
36 views

A multiplicative function satisfying $\lim_{p^m\to\infty} f(p^m)=0$ implies $\lim_{n\to\infty} f(n)=0$

Let $f$ be a multiplicative function satisfying $\lim_{p^m\to \infty} f(p^m)=0$. Show that $\lim_{n\to\infty} f(n)=0$. By unique factorization, we can write $n=\prod_{i=1}^k p_i^{\alpha_i}$, where $...
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0answers
29 views

What's about $ \sum_{n=1}^{\infty} \frac{ \mu\left( \sigma (n)\right)}{n^3} ,$ where $\mu(n)$ is Möbius function and $\sigma(n)=\sum_{d\mid n}d$?

Let $ \mu (n)$ the Möbius function and $ \sigma (n)$ the sum of divisors function, then the arithmetical function $g(n)= \frac{ \mu\left( \sigma (n)\right)}{n^3} $ isn't multiplicative since $gcd(2,...
1
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1answer
24 views

On the Density of Deficient Odd Numbers and Abundant Integers

Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) < 2x$, then $x$ is said to be deficient, while if $\sigma(x) > 2x$, $x$ is said to be abundant. (Of course, when $\sigma(x) ...
2
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1answer
65 views

On the proof of “The infinite series $\sum_{n=1}^{\infty} p_n^{-1}$ diverges”.

The following text is from the book Introduction to Analytic Number Theory by T. M. Apostol : Theorem 1.13 $ \ $ The infinite series $\sum_{n=1}^\infty 1/p_n$ diverges. Proof. The following ...
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1answer
44 views

Estimates of $\Omega_{\text{av}}(n)$

Ramanujan proved that the average number of distinct divisors of $x$ for $x$ on $[1,n], ~\omega_{\text{av}}(n),$ and the average number of divisors including repetitions, $\Omega_{\text{av}}(n),$ are ...
2
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0answers
58 views

Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...
5
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1answer
68 views

Combinations of four consecutive primes in the form $10n+1,10n+3,10n+7,10n+9$

Here $n$ is some natural number. For example, among the primes $< 1000$ I found four such combinations: \begin{array}( 11 & 13 & 17 & 19 \\ 101 & 103 & 107 & 109 \\ 191 &...
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0answers
49 views

Which values of $n$ is this inequality related to prime numbers true for?

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the $i$-...
0
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1answer
125 views

Does the Riemann Hypothesis for finite fields imply the original RH?

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ where $p$ is a prime. The zeta function, $\zeta(E, s)$ for $E$ is defined as $\zeta(E,s) = \dfrac{(1-\alpha p^{-s})(1-\beta p^{-s})}{(...
3
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1answer
39 views

irrationality of series of inverse of lcm

At the first time I'm dealing with this problem: prove that the series $\displaystyle\sum_{n=1}^{\infty} \frac{1}{d_{n}}$ is an irrational number, where $d_{n}=lcm(1,2,...,n)$. After some ...
10
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1answer
114 views

Equality involving Hasse zeta function of commutative ring finitely generated over $\mathbb{Z}$

Let $\mathbb{F}_q$ be a finite field consisting of $q$ elements. Imitating Riemann's zeta function$$\zeta(s) = \sum_{n = 1}^\infty {1\over{n^s}},$$define$$\zeta_{\mathbb{F}_q[t]}(s) = \sum_f {1\over{\...
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0answers
28 views

Convergence of the ratio of Gauss hypergeometric functions

Let $_2F_1(a,b;c;z)$ denote the Gauss hypergeometric function. Consider the following ratio for each $n\in\mathbb{N} $:$$n\cdot\frac{_2F_1(n+2,n+1;2;z)}{_2F_1(n+1,n;1;z)}.$$ Does this sequence ...
4
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0answers
90 views

Effect of 'Prime conspiracy' on the fact that prime numbers are the generators of integers [closed]

In Unexpected biases in the distribution of consecutive primes, the authors have discovered that prime numbers have decided preferences about the final digits of the primes that immediately follow ...
1
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1answer
48 views

Has limit $\frac{\sigma_0(n)\sigma_2(n)}{(\sigma(n))^2H_n},$ where $H_n$ is the nth harmonic number?

By specialization of an inequality I can write $$2 \sum_{k=1}^{n-1} \frac{1}{d_{k}} \sum_{l=k+1}^{n} \frac{1}{d_{l}}\leq 2\frac{\sigma_0(n)-1}{\sigma_0(n)}\cdot \left( \frac{\sigma(n)}{n} \right)^2, $...
2
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1answer
39 views

An entire Dirichlet series

Let $\{a(n)\}_{n\in\mathbb N}$ be a sequence of real number, suh that for any $C\in \mathbb{R}$ we have $$a(n)\ll_{C}n^{C}$$ My question : is how we can prove that the Dirichlet series $$\sum_{n=1}^{\...
0
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1answer
38 views

How to prove that $\sum_{d|n}d^{-\varepsilon}\leq C(\varepsilon)n^{\varepsilon}$

I wanna prove that for any $\varepsilon>0,$ there is a constant $C(\varepsilon )$ such that $$\sum_{d|n}d^{-\varepsilon}\leq C(\varepsilon)n^{\varepsilon}$$ but I do not know where I have to ...
1
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1answer
24 views

Are all Dirichlet coefficients of any element of the Selberg class necessarily algebraic?

The title says it all: do we know at least one element of the Selberg class having at least one transcendental coefficient in its development in a Dirichlet series for $\Re(s)>1$? Or are such ...
1
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1answer
50 views

What's about $\sum_{k=1}^{n-1} p_{k} \sum_{l=k+1}^{n} p_{l}$ for prime numbers?

By specialization of this formula, here in PROBLEMA 36, page 453 (in spanish), taking $\frac{1}{x_i}$ as the ith prime number we've (with at least two summands) $$ \left( \sum_{k=1}^{n} p_{k} \...
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0answers
36 views

Zeros of the second derivative of the modular $j$-function

I am interested in the zeros of $j''(z)$, where $j:\mathbb{H}\rightarrow\mathbb{C}$ is the classic modular function. Specifically I am interested in knowing if the zeros of $j''$ are algebraic over $\...
1
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2answers
944 views

Is it something new?

$W(n)$ is the function that counts number of distinct prime divisors of $n$. I have been able to prove for any $m$ consecutive integers starting with $1+a$ with the condition $a\leq (m^2-4m)/4$ , ...
2
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0answers
31 views
0
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0answers
11 views

Simultaneous degree two residues

Given integer $q$ is there many pairs of $x,y\in\Bbb Z$ with $q^{5/6}<x,y<2q^{5/6}$ such that $$q^{4/6}\quad<\quad x^2\bmod q,\quad xy \bmod q,\quad y^2\bmod q\quad<\quad 2q^{4/6}$$ holds?...
0
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0answers
31 views

If $\forall \varepsilon>0\;\;\;\;a(n)\leq C(\varepsilon)\; n^{r+\varepsilon} $ then $\sum_{n=1}^{\infty}\frac{a(n)}{n^s}$ converge

Let $\{a(n)\}_{n\in\mathbb{N}}$ be a sequence of real numbers. I tried to prove that if $$\forall \varepsilon>0\;\;\;\;a(n)\leq C(\varepsilon)\; n^{r+\varepsilon} $$where $C(\varepsilon)$ is a ...
0
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0answers
30 views

Weighted Q-binomial Coefficients

A possible identity popped up in a project for college, and if features q-binomial coefficient, which can be interpreted as the generating function for the number of Ferrer's boards fitting into a $k\...
1
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1answer
70 views

Analytic Number Theory: Problem in Bertrand’s postulate

I am trying to learn Bertrand’s postulate. I can not understand two steps Why $\displaystyle\sum_{n \leq x}\log n=\sum_{e \leq x} \psi\left(\frac{x}{e}\right)$, where $\psi(x)=\displaystyle\sum_{p^\...
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0answers
30 views

Upper Bound on Li's criterion

Background: Bombieri and Lagarias showed that a function $f$ with roots $\rho=x+iy$ satisfies has all its roots lying on $x=\frac12$ if and only if $$\lambda_n :=\sum_\rho 1-\left(1-\frac{1}{\rho}\...
4
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0answers
82 views

Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
2
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1answer
75 views

Asymptotic expression for sum of first n prime numbers?

Is one known? If not, what are the best known bounds? Is there reason to think that an asymptotic expression is beyond current methods if none exists?
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31 views

$\eta(s)+\eta(1-s)=F(s)-G(s)$ and roots of $F(s),G(s)$ are on the critical line

Wusheng Zhu in 2012 uploaded to arxiv.org an interesting preprint titled "Riemann Zeta Function Expressed as the Di fference of Two Symmetrized Factorials Whose Zeros All Have Real Part of 1/2" (arxiv:...
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0answers
32 views

Inequality of the Logarithmic Derivatives of a Sequence of Hypergeometric Functions

For brevity's sake, define the following sequence of (Gaussian) hypergeometric functions for each $n\in\mathbb{N}$: $$f_n(z)={}_2 F_1(-n,-(n-1);2;z).$$ I wish to show that the logarithmic derivatives ...
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0answers
10 views

Relation between support of a function and that of its DFT

Let $f : \mathbb{Z}_N \to \mathbb{C}$, let $\zeta_N$ be a primitive $N$-th root of unity and let $\hat{f} : \mathbb{Z}_N \to \mathbb{C}$ be the DFT of $f$ given by $\hat{f}(m) = \sum_{n \in \mathbb{Z}...
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0answers
24 views

Evaluating Certain $L$-functions

I have found a systematic way to find the exact value of the $L$-series $$L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$ for $s$ a positive even integer if $\chi(-1)=1$ and $s$ odd and positive if $...
2
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1answer
82 views

How do you refute these conjectures that seem imply contradictory statements?

I've formulated two conjectures that seems to imply a strong result when are combined with well known equivalences of the Riemann hypothesis, and I would like to know how get a disproof of such ...
4
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1answer
52 views

Context of this problem: $\sum_{n\,\text{odd}} (-1)^{\frac{n-1}2}\frac{\log n}{\sqrt{n}} /\sum_{n\,\text{odd}}(-1)^{\frac{n-1}{2}}\frac{1}{\sqrt{n}}$ [duplicate]

I remember seeing this somewhere a while ago - I'd given it a go but it was - and still is - beyond my capabilities. The problem came with the tag: "requires knowledge of analytic number theory". I am ...
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0answers
27 views

Questions about totient function

See this image (https://en.m.wikipedia.org/wiki/Euler%27s_totient_function#/media/File%3AEulerPhi.svg) from wikipedia. I can make out two lines that have a high density of values along them. The top ...
0
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1answer
29 views

Residue of Rankin-Selberg Dirichlet series

Let $f\in S_k(N,\chi)$ be a cusp forms, and let $R_f(s)=\sum_{n=1}^{\infty}\frac{a(n)^2}{n^s}$ the Rankin-Selberg Dirichlet series then $R_f(s)$ hase a pole at $s=k.$ Can someone suggest to me a ...
0
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0answers
41 views

Natural numbers, divisors, primes and their generalized means

Let div, nat and pri the finite sequences given in increasing order for an integer $n\geq 1$ of its divisors $1=d_1<d_2<\ldots d_{\sigma_0(n)}=n$, the first $n$ natural numbers, and the first $n$...
3
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0answers
19 views

A function related to divisior counting function

Let $d(n)$ be the divisor function. Let $d_{2}(n)=d(d(n))$, $d_{3}(n)=d(d(d(n)))$, $d_{4}(n)=d(d(d(d(n))))$ and so on... We're gonna define $f(n)$, the smallest number satisfies $d_{f(n)}(n)=2$. For ...
1
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1answer
34 views

$\sigma _{0}(n)=\sigma _{0}(n+1)$ will occur infinitely often. [closed]

In 1984, Roger Heath-Brown proved that will occur $\sigma _{0}(n)=\sigma _{0}(n+1)$ infinitely often. How did he prove that? I couldn't find the paper on the internet.
2
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2answers
54 views

What is the asymptotic behaviour of $\sum_{p_k\leq x}kp_k$, where $p_k$ is the kth prime number?

I would like to study the asymptotic behaviour of this sequence A014285, see as OEIS, that seems has few references and a good behaviour (see the sequence as graph) $$\sum_{k=1}^nkp_k,$$ where $p_k$ ...
0
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0answers
58 views

On the asymptotic limit of the divisor function.

It is known that $$ \limsup_{k \to \infty} \frac{\sigma(N_k)}{e^{\gamma}N_k \log \log N_k} = \frac{6}{\pi^2}$$ Where $N_k$ is the $k$-th primorial, $\sigma$ is the divisor function and $\gamma$ is ...
2
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0answers
32 views

Finite sequence created by reducing $n$ with each prime under $n$ ends in $0$?

Given $n$ a fixed integer we constuct the following sequence: $a_0=n$, $a_i=\lfloor \frac{a_{i-1}(p_i-1)}{p_i}\rfloor$. For what values of $n$ do we have $a_{\pi(n)}=0$? Computer calculation shows ...
2
votes
0answers
27 views

A linearly uniform but quadratically non-uniform set

I have been working on this problem for a while but have no clue at all. Fix a smooth cutoff function $\varphi: \mathbb R/\mathbb Z\rightarrow [0,1]$ supported on $[−\varepsilon-\delta, \...
0
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1answer
51 views

Bound a complex exponential sum when we can only estimate its argument

Suppose we have a function $f : \mathbb{N_0} \to \mathbb{R}$ for which we can give an estimate of its values, and say its values $f(n)$ are roughly uniformly distributed for $n$ in some range $[1,N]$. ...