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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Regarding similarly ordered fractions in farey sequences

This question is from Apostol modular functions and Dirichlet series in number theory. It is related to this problem - When are two neighbouring fractions in Farey sequence are similarly ordered Apostol in 2 nd part of this exercise asks to prove…
Avenger
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Regarding expressing Lambert series in terms of Dirichlet Convolution

I am studying Lambert Series . It's definition says a series of the form $\sum_{n=1}^\infty \frac { f(n) x^n } { 1 - x^n } $ = $\sum_{n=1}^\infty F(n) x^n $ , where $F(n) = \sum_{d|n} f(d) $ . I can think about LHS of defination equal to…
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Regarding proving a result related to $\alpha$- multiplicative function from exercises of Tom M Apostol

I am trying exercises of Tom M Apostol Modular functions and Dirichlet series in number theory of Chapter-6 and I could not think about this problem. This problem uses concepts introduced in beginning of 1st problem but note that I have doubt only…
Avenger
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Problem of Ch -2 Of Apostol Modular functions and Dirichlet series in number theory

I am trying exercises of Tom Apostol Modular functions and Dirichlet Series in Number Theory and I cannot think about this problem of Chapter 2 . Problem 2 - For each prime p the number of solutions, distinct mod $p^r$ , of all possible…
Avenger
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The Proofs of Prime Number Theorems

I had decided to study and understand the proof of prime number theorem. I had tried to approximate the value of $\frac{\pi(x)}{x}$ and have seen some videos which gives the intuition about the approximation. So I want to ask for references for a…
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Bounding Integral Involving Riemann Zeta

For $2 \le T \le x$, can we bound $$\int_1^{1+\frac{1}{log x}}\frac{x^s}{s}\zeta(\sigma + iT) \, \mathrm{d}\sigma \ll \frac{x \log^3 x}{T}$$ Question Background We get the expression for $Z(s)$ by writing the Dirichlet series corresponding to…
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A sum for the $q$-expansion of Eisenstein series of weight $2.$

Let $N>1$ be a positive integer, let $(c_v,d_v)$ be positive integers such that $gcd(c_v,d_v,N)=1,$ and $0 \leq c_v \leq N-1.$ I have seen it claimed that $$\dfrac{1}{N^2}\sum_{c<0} \sum_{d \in \mathbb{Z}}…
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Range of Dirichlet $L$ Function

I was reading A Course in Arithmetic by Serre and on page 74 he defined the log of the Dirichlet L functions. (Previously, he showed that $L(1,\chi)\neq 0$ when $\chi\neq 1$). Based on that fact he showed that $log(L(1,\chi))$ is bounded as…
Jhon Doe
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Behavior of $ f_p$ under generator $S\tau = ( -1 / \tau )$

I am self studying Tom M Apostol Modular functions and Dirichlet series in number theory and I could not think about a step in the proof of the theorem 4.6 in Chapter 4 which is If $f$ is automorphic under $\Gamma $ and if $p$ is a prime then, …
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Estimate for $n$-th Prime.

I am a bit stuck on how to derive the following estimate for the $n$-th prime. $$p_n = n \log n + n \log \log n + \mathcal{O} (n)$$ (We are given that $\pi (x) = Li(x) + \mathcal{O}(\exp(-a \sqrt{\log x}))$ for some constant $a$.) I am a bit stuck,…
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For what primes $p$ does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?

this is a question from a book I'm struggling with, please could you provide a clear proof For what primes p does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically? kind thanks
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Why is the Ruelle zeta function called a zeta function?

Let $f \colon X \to X$ be a topological dynamical system. Write $|\operatorname{Fix}(f^n)|$ for the number of fixed points of the $n$-fold composition of $f$. Then the Ruelle zeta function is defined as $$\zeta(s) = \exp \bigg( \sum_{n \geq 1}…
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Convergence Locally in Uniform Convergence

As far as I understand when we say uniformly, it means we are indicating an area where the function converges, isn't is always local when something converges uniformly? Please explain what does locally convergence mean. This question is related to…
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Confusion about big O notation in approximation to zeta function

My notes state that if $s=\sigma+it$ with $\sigma>1$ and $t\in\mathbb{R}$, then for any $x\in\mathbb{N}$ we have $$\zeta(s)=\sum_{n\leq x} \frac{1}{n^s} + \frac{x^{1-s}}{s-1} + O\left(\frac{|s|}{x^{\sigma}}\right)$$ and in particular that if…
AlephNull
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