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

votes

Recently, I stumbled upon what I believe to be a new representation of $\zeta(\sigma+i t)^b$ by chance, and thus have no proof of it, and I am wondering if it is possible to prove.
Let $\eta (s) = \zeta (s) (1-2^{1-s})$ denote the Dirichlet eta…

KStarGamer

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I'm trying to prove the infinitude of primes as follows:
Consider the following partial sum :
$$S(p)=\sum_{n=2}^p\sin^2\left(\frac{π\Gamma(n)}{2n}\right)$$
The summand is zero for non-primes greater than 5 , and finite and non-decreasing for primes…

bambi

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Does there exist positive rational $s$ for which the Riemann Zeta function $\zeta(s) \in N$ or equivalently, does there exist finite positive integers $\ell,m$ and $n$ such that $$\zeta\left(1+\dfrac{\ell}{m}\right) = n$$
Update: 16-Apr-2021: New…

Nilotpal Sinha

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A positive integer is $B$-smooth if and only if all of its prime divisors are less than or equal to a positive real $B$. For example, the $3$-smooth integers are of the form $2^{a} 3^{b}$ with non-negative exponents $a$ and $b$, and those integers…

user02138

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votes

We have the following evaluations:
$$\begin{aligned}
&\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\
&\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = \frac{1}{3}\,\zeta(2)\\
&\sum_{n=1}^\infty \frac{1}{n^3\,\binom {2n}n} =…

Tito Piezas III

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votes

Let $\pi(x)$ denote the Prime Counting Function.
One observes that, $\pi(6) \mid 6$, $\pi(8) \mid 8$. Does $\pi(x) \mid x$ for only finitely many $x$, or is this fact true for infinitely many $x$.

user9413

votes

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written
$$
\pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1}
$$
with $ \operatorname{R}(z) = \sum_{n=1}^{\infty} \frac{ \mu (n)}{n}…

draks ...

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I would like to evaluate the sum
$$
\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)
$$
Where $\operatorname{Si}$ is the sine integral, defined as:
$$\operatorname{Si}(x) := \int_0^x \frac{\sin t}{t}\, dt$$
I found that the…

Argon

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The least common multiple of $1,2,\dotsc,n$ is $[1,2,\dotsc,n]$, then
$$\lim_{n\to\infty}\sqrt[n]{[1,2,\dotsc,n]}=e$$
we can show this by prime number theorem, but I don't know how to start
I had learnt that it seems we can find the proposition…

Clin

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So I've been thinking about the analytic class number formula lately, and class numbers in general and I'm trying to develop a good intuition for them. My basic question, which may be too general/difficult is: $\textbf{What forces the class number…

Dylan Yott

- 6,851
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- 37

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I'd like to calculate, or find a reasonable estimate for, the Mertens-like product
$$\prod_{2

Charles

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I know a little bit about basic number theory, much about algebra/analysis, I've read most of Niven & Zuckerman's "Introduction to the theory of numbers" (first 5 chapters), but nothing about analytic number theory. I'd like to know if there would…

Patrick Da Silva

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I was looking for a closed form but it seemed too difficult. Now I'm seeking help to simplify this sum. The 50 bounty points or more will be awarded for any meaningful simplification of this sum.
I found this function that has very interesting…

Fred Yang

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Legendre's conjecture is that there exists a prime number between $n^2$ and $(n+1)^2$. This has been shown to be very likely using computers, but this is merely a heuristic. I have read that if this conjecture is true, the biggest gap between two…

Adam

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I've been studying modular forms this semester and we did a lot of calculations of $L$-functions, e.g. $L$-functions of Dirichlet-characters and $L$-functions of cusp-forms.
But I somehow don't see, why they are considered a big deal. To me it…

Steven

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