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

Analytic number theory is a branch of mathematics that uses techniques in real and complex analysis to study the integers (including the primes).

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

Analytic number theory is a branch of mathematics that uses techniques in real and complex analysis to study the integers (including the primes).

3532 questions

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I'm having trouble verifying an inequality that is claimed in "PRIMES is in P" (Annals of Mathematics, 160 (2004), 781-793).
I'll state it in such a way that it can be understood without the paper, but for the curious it is in the middle of the…

Adam Smith

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I am unable to derive this result assuming prime number theorem. Can someone please tell how to do it.
Edit -> Here is a proof from stackexchange >but I couldn't think how last line is true. Can someone please tell how it's true
If $d_n =…

Avenger

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I read this proof of the prime number theorem:
PRIME NUMBER THEOREM:
$ψ\left(x\right)-x=O\left(xe^{-c\sqrt{\log \left(x\right)}}\right)$ for some effective $c\in\mathbb{R}_{+}$
PROOF:
$ψ\left(x\right)-x=O\left(∑_{_{|γ|\leq…

iwonnaRoch

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While trying problems from above mentioned book I am unable to think about how to prove the question which I am writing below.
Question is ->If limit x->$\infty \frac {π(x) } {x/log(x) } $ = $\alpha$
Then show that $\sum_{p\leq x} 1/p = \alpha…

Avenger

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Let $Λ$ be the von-Mangoldt function. then
What is the estimate for the sum $\sum_{1\leq x\leq n}\Lambda(x)^{4}$?
Is this $\sum_{1\leq x\leq n}\Lambda(x)^{4}\sim n\log^3n$
also what can we say about this when $x\neq y$?
$ \sum_{1\leq x, y\leq…

Math123

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I am self studying analytic number theory from lecture notes of Peter Bruin and Sanders Dahmen.
I have a doubt in proof when the authors writes the identities involving order of zeroes. I am posting the image highlighting the the part I have doubt.…

Avenger

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It is well know that the The sum of the reciprocals of all primes diverges. How to prove that The sum of the reciprocals of Chen primes converges.

Neil hawking

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While self studying analytic number theory from Tom M Apostol modular functions and Dirichlet series in number theory I am unable to think about an argument which Apostol doesn't proves but uses it in Theorem 1.14 of chapter - elliptic…

Avenger

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I'm working through Introduction to Analytic Number Theory by Tom Apostol. I've come across this question (Chapter 4, Exercise 17),
Given an integer $n > 1$ with two factorizations $n = \prod_{i=1}^r p_i$ and $n = \prod_{i=1}^t q_i$, where the…

eeen

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While studying chapter partitions from Apostol introduction to analytic number theory I have a doubt on page number 311 .
Apostol defines inverse of partition function $\prod_{m=1}^{\infty} 1 - x^m $ = 1+ $\sum_{n=1}^{\infty} a(n) x^n $ .
Then this…

Avenger

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Need help solving the integral $$\int_{c-i\infty}^{c+i\infty} \frac{P(s)\cdot x^s}{s^2}ds$$, where $P(s)$ is the prime zeta function.
If the denominator was simply $s$, then this integral would nicely evaluate to $\pi(x)$ (multiplying the last term…

Pedro Mariz

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The Artin $L$-series for Abelian extensions are known to coincide with Hecke $L$-series, which in particular implies that if $E/K$ is a Abelian extensions, and $\chi$ is a non-trivial simple character of $\textrm{Gal}(E/K)$, then $L(E/K,\chi,s)$…

Heinrich Wagner

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Suppose I already know that Mertens' third theorem holds in the form
$$\prod_{p\leq x} \left(1 - \frac{1}{p}\right) =
\frac{C}{\ln x} + O\left(\frac{1}{\ln^2 x}\right)$$
for some constant $C$. I wonder what the easiest way is to show that $C =…

Manuel Eberl

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I am studying Analytic number theory from Tom M Apostol Introduction to analytic number theory and I have doubt in the proof given in the book.
Dirichlet gives two proof of formula
1 st has error term $O(x)$ and I don't have any doubt in it.
I…

Avenger

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I was trying a problem from Ch -1( Elliptic Functions problem no. 15) of book Modular functions and Dirichlet series in number theory whose Statement is this.
Image-
I have no idea on how to solve this problem, please give some hints. It is after…

Avenger

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