Questions tagged [analytic-number-theory]
Questions on the use of the methods of real/complex analysis in the study of number theory.
572
questions
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How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $?
Note the $ p < x $ in the sum stands for all primes less than $ x $. I know that for $ s=1 $,
$$ \sum_{p<x} \frac{1}{p} \sim \ln \ln x , $$
and for $ \mathrm{Re}(s) > 1 $, the partial sums ...
- 152k
22
votes
5
answers
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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. ...
- 321
5
votes
2
answers
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Is there a way to show that $\sqrt{p_{n}} < n$?
Is there a way to show that $\sqrt{p_{n}} < n$?
In this article, I show that $f_{2}(x)=\frac{x}{ln(x)} - \sqrt{x}$ is ascending, for $\forall x\geq e^{2}$. As a result, $\forall n \geq 3$ $$\frac{...
- 17.1k
10
votes
2
answers
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Bounds for $\zeta$ function on the $1$-line
I was going over my notes from a class on analytical number theory and we use a bound for the $\zeta$ function on the $1$ line as $\vert \zeta(1+it) \vert \leq \log(\vert t \vert) + \mathcal{O}(1)$ ...
- 301
0
votes
1
answer
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General solution of a linear congruence $\,ax\equiv b\pmod m$ [closed]
I am self studying number theory from Introduction to Analytic number theory by Apostol.
I have a doubt in one argument of proof of Theorem 5.14 and I am posting its image.
My doubt is in 2nd ...
user775699
21
votes
3
answers
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Calculate $\sum\limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$
Calculate $$\sum \limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$$
I use software to complete the series is $\frac{2}{27} \left(18+\sqrt{3} \pi \right)$
I have no idea about it. :|
- 1,190
29
votes
4
answers
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What is the analytic continuation of the Riemann Zeta Function
I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation. What is this other ...
- 17.5k
22
votes
3
answers
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How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?
I saw reference to this result of Chebyshev's:
$$\sum_{p\leq n} \frac{\log p}{p} \sim \log n \text{ as }n \to \infty,$$
and its relation to the Prime Number Theorem. I'm looking into an information-...
- 3,774
13
votes
1
answer
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Limit inferior of the quotient of two consecutive primes
I have recently read an article about the prime number theorem, in which Mathematicians Erdos and Selberg had claimed that proving $\lim \frac{p_n}{p_{n+1}}=1$, where $p_k$ is the $k$th prime, is a ...
- 2,530
8
votes
2
answers
2k
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Asymptotic formula for $\sum_{n\leq x}\mu(n)[x/n]^2$ and the Totient summatory function $\sum_{n\leq x} \phi(n)$
I would like to show (for $x \ge 2$) that $$\sum_{n \le x}\mu(n)\left[\frac{x}{n}\right]^2 = \frac{x^2}{\zeta(2)} + O(x \log(x)).$$
I already have the identity $$\sum_{n \le x}\mu(n)\left[\frac{x}{n}\...
- 12.5k
4
votes
3
answers
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How to prove $\sum_{d|n} {\tau}^3(d)=\left(\sum_{d|n}{\tau}(d)\right)^2$
For every positive integer $d$, we let $\tau\left(d\right)$ be the number of positive divisors of $d$.
Prove that
\begin{align}
\sum_{d|n} \tau^3(d)
= \left(\sum_{d|n} \tau (d)\right)^2
\end{align}
...
- 2,879
2
votes
3
answers
242
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Is the ratio of consecutive Bernoulli polynomials uniformly bounded [duplicate]
When investigating a certain kind of Stirling's approximation of the Gamma function error terms occur such as
\begin{equation}
E(s)=\frac{1}{s}\sum_{j=1}^\infty B_{j+1}(a)\frac{(-1)^{j+1}}{j(j+1)s^{j-...
- 317
16
votes
3
answers
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Are there any Combinatoric proofs of Bertrand's postulate?
I feel like there must exist a combinatoric proof of a theorem like: There is a prime between $n$ and $2n$, or $p$ and $p^2$ or anything similar to this stronger than there is a prime between $p$ and $...
- 12.5k
15
votes
2
answers
5k
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Sum of reciprocal prime numbers [duplicate]
How can the following equation be proven?
$$ \forall n > 2 : \sum_{p \le n}{\frac1{p}} = C + \ln\ln n + O\left(\frac1{\ln n}\right), $$ where $p$ is a prime number.
It's not homework; I just ...
- 253
14
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3
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Lower bound for $\phi(n)$: Is $n/5 < \phi (n) < n$ for all $n > 1$?
Is it true that :
$\frac {n}{5} < \phi (n) < n$ for all $n > 1$
where $\phi (n)$ is Euler's totient function .
Since $\phi(n)$ has maximum value when $n$ is a prime it follows that maximum ...
- 12.9k