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

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10
votes
1answer
117 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{\...
0
votes
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
votes
0answers
100 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
vote
1answer
50 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
votes
1answer
44 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
votes
1answer
42 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
vote
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
vote
1answer
53 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} \...
2
votes
0answers
38 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
vote
2answers
945 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
votes
0answers
31 views
0
votes
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
votes
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
votes
0answers
31 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
vote
1answer
71 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^\...
0
votes
0answers
32 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
votes
0answers
84 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
votes
2answers
103 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?
0
votes
0answers
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:...
0
votes
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 ...
0
votes
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}...
1
vote
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
votes
1answer
83 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
votes
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 ...
0
votes
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
votes
1answer
30 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
votes
0answers
42 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
votes
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
vote
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
votes
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
votes
0answers
59 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
votes
0answers
33 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
votes
1answer
55 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 $[1,N]$. ...
1
vote
0answers
39 views

Meaning of “Form” in mathematics

There are a number of occasions in the mathematics where the word "form" is used: Modular form Bilinear form Quadratic form ... It seems that form is a special kind of function. But I cannot ...
1
vote
0answers
18 views

Can you discuss $\limsup_{n\to\infty}\frac{g_n}{\log^2p_n}\cdot\frac{\sigma(K_n)}{K_n\cdot\log\log K_n}$, where $g_n=p_{n+1}-p_n$ and $K_n\to\infty$?

Let $p_n$ the nth prime number, then we know that the nth gap is $g_n=p_{n+1}-p_n$. We define for $n>1$, $C_n$ as the set of integers such that $gcd(k,p_{n+1})=gcd(k,p_{n})=1$, this is $\{1\leq k &...
0
votes
0answers
37 views

Who extended the Euler Product Formula to all real $s>1$?

I believe Euler discovered this identity but only wrote them for particular values of $s$, then Chebychev extended to real $s>1$. However, I read in the book Riemann's Zeta Function, H.M. Edwards ...
0
votes
0answers
46 views

How do you justify these statements from $\psi(x)=x-\sum_{\rho:\zeta(\rho)=0}\frac{x^\rho}{\rho}-\log 2\pi-\frac{1}{2}\log(1-x^{-2})$?

In the second paragraph of page 343 of this paper, NOTICES OF THE AMS Vol. 50, No. 3 the authors wrote the following concerning to the sum $\sum_{n\leq x}\Lambda(n)$, where $\Lambda(n)$ is von Mangold ...
3
votes
1answer
52 views

Can you give an understandable strategy to compute the asymptotic behaviour of $\sum_{2\leq n\leq x}\frac{\Lambda(n)Li(n)}{n}$?

First I am looking a discussion about if the statements that I've deduced will find a good asymptotic formula, what of them you can discard as non useful,or one better than other, and ... second my ...
1
vote
1answer
48 views

Exploring the Dirichlet series of the sum of remainder function

I wolud like to learn and understand more some basic facts about Dirichlet series, for wich I want explore the following function, that is called the sum of remainders function, A004125 as Sloane's ...
0
votes
0answers
20 views

It is possible an application of Shapiro's Tauberian theorem for $\sum_{n\leq x}\frac{|M(n)|^{1+\alpha}}{\pi(n)^{1+\beta}}\left[\frac{x}{n}\right]$?

I would like to know if I can find some $\alpha,\beta\geq 0$ such that defining the sequence $a(1)=1$ and for $n>1$ as $$a(n)=\frac{|M(n)|^{1+\alpha}}{\pi(n)^{1+\beta}},$$ where $M(n)=\sum_{k\leq n}...
1
vote
1answer
65 views

Can you provide us an understandable and detailed explanation of the relationship between prime numbers and equidistibution theory?

In this site, in 2. the user that answered the question wrote "the prime number theorem is an equidistribution theorem, and the Riemann Hypothesis an optimal discrepancy bound. By bounding this ...
2
votes
1answer
56 views

Evaluating $\lim_{x\to\infty}\frac{1}{x}\int_2^x M(t)\cdot f'(t)dt$, where $M(x)$ is Mertens functions

Let $\mu(n)$ the Möbius function. I know that combining Abel summation formula, the Prime Number Theorem and l'Hôpital's rule I can deduce $$\lim_{x\to\infty}\frac{1}{x}\sum_{2\leq n\leq x}\mu(n)\cdot\...
0
votes
1answer
46 views

Vaughan's identity, a didactic example

I know that Vaughan's identity is one of the methods used in analityc number theory, I would like see an example, I say a simple example of application of this theorem for encourage to study the ...
0
votes
1answer
60 views

Poisson summation in analytic number theory, an example

I've read a theorem of a course of analytical methods in theory of numbers, and I want to ask to clarify it, since I know this theorem from other context. The theorem is Poisson summation formula (I ...
0
votes
0answers
34 views

Additive combinatorics: Switching $\mathbb{Z}/N\mathbb{Z}$ with $\mathbb{Z}$

For a positive integer $N$, denote $\mathbb{Z}_N = \mathbb{Z}/N\mathbb{Z}$. Now let $F$ be a field and let $f_1, \ldots, f_s : \mathbb{Z}_N \to F$. Can we find functions $\tilde{f_1}, \ldots, \tilde{...
1
vote
1answer
48 views

A didactic example of logarithmic measure

I've read that two mathematicians, in the recent past, studied limits as $X$ tends to infinite, like as $$\frac{1}{\log X}\sum_{x\leq X:\pi(x)<\int_{2}^x\frac{dt}{\log t}}\frac{1}{x},$$ where $\pi(...
1
vote
0answers
54 views

Can you estimate the difference of primes between numerator and denominator?

Let $p_n$ the nth twin prime, it is $p_n$ is a prime number and $2+p_n$ is also a prime. It is well know that Brun's theorem states (unconditionally) that $$\mathcal{B}=\sum_{n\geq 1}\left(\frac{1}{...
0
votes
0answers
22 views

Number of ways, modulo a prime $p$, to write $n$ as a sum $n = x_1^k + x_2^k + \cdots + x_s^k$

Removing the restriction on $p$, this is known as Waring's problem. The circle method has been used successfully to tackle this. Using analysis, nice estimates can be given. I wonder what analytic ...
1
vote
0answers
37 views

On an inequality involving primorial numbers.

Let $N_k$ denote the $k-th$ primorial number. That is, the product of the first $k$ primes and $\phi(n)$ be the Euler totient function. How can one show that there exists a constant $\theta>1$ ...