# Questions tagged [analytic-number-theory]

A beautiful blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.

1,669
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### Number of matrices with bounded products of rows and columns

Fix an integer $d \geq 2$ and for every real number $x$ let $M_d(x)$ be number of $d \times d$ matrices $(a_{ij})$ satisfying: every $a_{ij}$ is a positive integer, the product of every row does not ...

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### Asymptotic size for the number of terms not exceeding $n$ in the class $r$ for a classification of the type Erdös-Selfridge for square-free integers

It is possible to define a classification similar than the Erd?s-Selfridge classification of primes for different sequences. Please ee [1], section A18 and the references cited in this book. Because ...

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**1**answer

123 views

### Books on complex analysis for self learning that includes the Riemann zeta function?

I am searching for an introductory book in the field of complex analysis for self learning, that would contain the following:
Analytic number theory : the connection between complex analysis and
...

**2**

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**1**answer

129 views

### An asymptotic formula for a sum involving powers of floor functions

Let $\theta \geq 0$ and consider the sum $$\sum_{n \leq x} \left\lfloor \frac{x}{n} \right\rfloor^{-\theta}.$$.
I have seen the claim that there is a constant $c(\theta)$ (depending on $\theta$!) ...

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### Convergence with the recurrence $T_{n+1}=T_n^2-T_n+\frac{n}{p_n}$

For each integer $n\geq 1$ I define the recurrence $$T_{n+1}=T_n^2-T_n+\frac{n}{p_n},$$
with $T_1=1$, where $p_k$ denotes the $k-th$ prime.
So multiplying by $(-1)^n$ and telecoping gives that for ...

**10**

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**1**answer

329 views

### turn $\pi/n$, move $1/n$ forward

start at the origin, first step number is 1.
turn $\pi/n$
move $1/n$ units forward
Angles are cumulative, so this procedure is equivalent (finitely)
to
$$
u(k):=\sum_{n=1}^{k} \frac{\exp(\pi i H_{n}...

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### Relatively prime polytope extension complexity

What is the extension complexity of the $0/1$ vertexed polytope in $2d$ dimensions with property that integer represented by first $d$ coordinates is coprime to integer represented by second $d$ ...

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101 views

### Prime numbers and sieving up to $q(x)=\log(x)(1+o(1))$

Let $x,z\in\mathbb{R}_{+}$
Let $P_z = \displaystyle{\small \prod_{\substack{p \leq z \\ \text{p prime}}} {\normalsize p}}$
Let $I(x, z)=\{k \in \mathbb{N} \, | \, k \leq x \text{ and } \gcd(k, P_z)=...

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78 views

### From Firoozbakht's conjecture to set interesting conjectures for sequences or series of primes

In this post we denote the $k-th$ prime number as $p_k$. I present two conjectures, the first about the asymptotic behaviour of a product and the other about an alternating series. I don't know if ...

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28 views

### On values of $n\geq 1$ satisfying that for all primorial $N_k$ less than $n$ the difference $n-N_k$ is a prime number

In this post I present a similar question that shows section A19 of [1], I was inspired in it to define my sequence and question. I am asking about it since I think that the problem that arises from ...

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**1**answer

242 views

### Reference request for Euler products in positive characteristic

Let $K$ be a global field with ring of integers $O$, and let $f$ some integer valued function whose domain is the set of ideals of $O$ (e.g. $f(I)=|O:I|$). Extending the ordinary definition from the ...

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34 views

### Bounds for the number of prime numbers less than the Euler's factor, the radical and the greatest prime factor, respectively, of an odd perfect number

As tell us the Wikipedia section dedicated to Odd perfect numbers (please, see also the related references if you need it), any perfect number has the form $$n=q^\alpha m^2$$
where the integer $\alpha\...

**4**

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72 views

### Is the ratio of a number to the variance of its divisors injective?

The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer whose variances are equal e,g. $v_{691} = v_{817}$. However I observed that for $n \le ...

**0**

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**1**answer

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### On the integral $I_s = \int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$-follow up question

This is a follow up on On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$
According to the answer that i got, $I_s$ is not known to converge for any real $s<1$. But suppose $I_s$ ...

**1**

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37 views

### Non-vanishing Taylor coefficients and Poincaré series

I'm studying these algebraic numbers in the table below and have succesfully shown what i wanted with the given values I had.
The table is found in the book "The 1-2-3 of Modular Forms" by Jan ...