Questions tagged [analytic-number-theory]

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).

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Inequality in "PRIMES is in P"

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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How to deduce a result assuming prime number theorem

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 =…
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Proof of the prime number theorem

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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Doubt in exercise 3.1.12 of Book Problems in analytic number theory by My Ram Murthy

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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Sum with the von-Mangoldt function: $\sum_{1\leq x\leq n}\Lambda(x)^4$

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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Doubt in proof of valence formula of entire modular forms

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.…
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The sum of the reciprocals of Chen primes converges

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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Regarding property of odd elliptic functions

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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Zeros of $f(\alpha) = \sum_{i=1}^r p_i^\alpha - \sum_{i=1}^t q_i^\alpha$ (Apostol ANT 4.17)

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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Doubt in partition function generated by reciprocal of generating function of p(n).

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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Line integral of prime zeta function

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…
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Artin conjecture for degree 1 representations

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 $L(E/K,\chi,s)$…
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Proving that the constant in Mertens' third theorem is $e^{-\gamma}$

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 =…
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Doubt in Proof of Dirichlet Asymptotic formula for partial sum of divisor function $d(n)$

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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Regarding proving a series result from Tom M Apostol Modular functions and Dirichlet series in number theory

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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