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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What is a zeta function?

In my readings, I've come across a wide variety of objects called zeta functions. For example, the Ihara zeta function, Igusa local zeta function, Hasse-Weil zeta function, etc. My question is simple: What makes something a zeta function? There are…
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Newman's "Natural proof"(Analytic) of Prime Number Theorem (1980)

I am trying to understand this short proof by newmann. I faced some problems while grasping this very proof. Please help me out. 1 . I am not clear, why in step (1)'s proof he says that from unique factorization and the absolute convergence of zeta…
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Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good feeling for which numbers are primes, but we do know…
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Evaluating $\sum\limits_{n=1}^{\infty} \frac{1}{n\operatorname{ GPF}(n)}$, where $\operatorname{ GPF}(n)$ is the greatest prime factor

$\operatorname{ GPF}(n)=$Greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. How to evaluate convergence/divergence/value of the sum $$\sum_{n=1}^{\infty} \frac{1}{n\operatorname{ GPF}(n)}\,?$$
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Upper bound on differences of consecutive zeta zeros

The average gap $\delta_n=|\gamma_{n+1}-\gamma_n|$ between consecutive zeros $(\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i)$ of Riemann's zeta function is $\frac{2\pi}{\log\gamma_n}.$ There are many papers giving lower bounds to $$ \limsup_n\…
23
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On prime factors of $n^2+1$

It is a well-known conjecture that there are infinitely many primes of the form $n^2+1$. However, there are weaker results that one can prove. For example, There are infinitely many positive integers $n$ such that $n^2+1$ has a prime divisor…
23
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What is the analytic continuation of the Riemann Zeta Function

I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation. What is this other function that they use explicitly given? Also if…
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On Dirichlet series and critical strips

(I'll keep this one short) Given a Dirichlet series $$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$ where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero real numbers), and assuming that $g(s)$ can be…
J. M. ain't a mathematician
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How many elements in a number field of a given norm?

Let $K$ be a number field, with ring of integers $\mathcal{O}_k$. For $x\in \mathcal{O}_K$, let $f(x) = |N_{K/\mathbb{Q}}(x)|$, the (usual) absolute value of the norm of $x$ over $\mathbb{Q}$. Dirichlet's unit theorem tells us, in a precise sense,…
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How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?

I saw reference to this result of Chebyshev's: $$\sum_{p\leq n} \frac{\log p}{p} \sim \log n \text{ as }n \to \infty,$$ and its relation to the Prime Number Theorem. I'm looking into an information-theory proof by Kontoyiannis I was wondering if…
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Main differences between analytic number theory and algebraic number theory

What are some of the big differences between analytic number theory and algebraic number theory? Well, maybe I saw too much of the similarities between those two subjects, while I don't see too much of analysis in analytic number theory.
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Motivation on how does complex analysis come to play in number theory?

I am not sure if this is a appropriate question. If it isn't, let me know and I'll delete it. $\textbf{Background}$ I am an undergraduate student and I'm very interested in number theory. I've tried to study alone some algebraic number theory and I…
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How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?

I mean the Laurent series at $s=1$. I want to do it by proving $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$, based on the integral formula given in Wikipedia. But I cannot solve this integral except by using…
Victor L
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Calculate $\sum\limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$

Calculate $$\sum \limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$$ I use software to complete the series is $\frac{2}{27} \left(18+\sqrt{3} \pi \right)$ I have no idea about it. :|
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Intervals that are free of primes

How can I prove that exists intervals as large as I want that are free of primes? I mean, $\forall \ k \in \mathbb{N}, \exists \ k$ consecutive positive integers none of which is a prime.