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# 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.

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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 , section A18 and the references cited in this book. Because ...
1answer
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### 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 ...
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### 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 $\theta$!) ...
0answers
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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 ...
1answer
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1answer
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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 $I_s$ ...
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### 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 ...

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