# Tagged Questions

Questions on the use of the methods of real/complex analysis in the study of number theory.

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### How hard is the proof of $\pi$ or $e$ being transcendental?

I understand that $\pi$ and $e$ are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
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### What is so interesting about the zeroes of the Riemann $\zeta$ function?

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$ The ...
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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-...
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### Using the Brun Sieve to show very weak approximation to twin prime conjecture

I recently stumbled across MIT OCW for analytic number theory. As a budding number theorist, my ears perked up and I looked through some of the material haphazardly. I don't really know much about ...
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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.
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### A Conjecture of Schinzel and Sierpinski

Melvyn Nathanson, in his book Methods in Number Theory (Chapter 8: Prime Number's) states the following: A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can ...
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### Simplify the sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes $\le N^2$

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 ...
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### A good reference to begin analytic number theory

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 ...
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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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### Logarithmic derivative of Riemann Zeta function

Given the logarithmic derivative of the zeta function $\dfrac{\zeta^\prime (s)}{\zeta(s)}$ how does it behave near $s=1$? I mean if for $s=1$ the Laurent series for the logarithmic derivative becomes ...
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### Why are $L$-functions a big deal?

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, ...
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### Series of the totient function

Good morning, I wonder if : $$\sum_{n} \frac{(-1)^n}{\varphi (n)}$$ converges or not. where $\varphi (n)$ is the Euler function. Do you have any idea ?
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Let $$G(n)=\begin{cases}1 &\text{if }n \text{ is a prime }\equiv 3\bmod17\\0&\text{otherwise}\end{cases}$$ And let $$P(n)=\begin{cases}1 &\text{if }n \text{ is a prime }\\0&\text{... 6answers 315 views ### Is  \sin: \mathbb{N} \to \mathbb{R} injective? I was trying to show that \sin(x) is non-zero for integers x other than zero and I thought that this result might emerge as a corollary if I managed to show that the result in question is true. ... 2answers 497 views ### least common multiple \lim\sqrt[n]{[1,2,\dotsc,n]}=e The least common multiple of 1,2,\dotsc,n is [1,2,\dotsc,n], then$$\lim_{n\to\infty}\sqrt[n]{[1,2,\dotsc,n]}=e$$we can show this by prime number theorem, but I don't know how to start I ... 1answer 777 views ### Values of hypergeometric functions Let _pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;c) denote the generalized hypergeometric function. Let A \subset \mathbb R be the set of all values of \ _pF_q(\cdot) at rational points a_i,b_j,c\in \... 1answer 319 views ### Chebyshev: Proof \prod \limits_{p \leq 2k}{\;} p > 2^k How do I prove the following:$$\prod_{p \leq 2k} \; p > 2^k \text{ with } p \in \mathbb{P}$$I tried induction, but I didn't know how to go on because I don't have a look at all numbers. ... 3answers 541 views ### Rate of divergence for the series \sum |\sin(n\theta) / n| In the following we consider the series$$ S(N;\theta)= \sum_{n = 1}^{N} \left| \frac{\sin n\theta}{n} \right| $$parametrized by \theta. It is well known that this series (taking the limit N\to\... 1answer 521 views ### What is the probability that some number of the form 10223\cdot 2^n+1 is prime? I (David Speyer) took the liberty of doing a fairly major rewrite of this question. I hope I haven't gone too far, but I think there is an interesting question hiding here. Sierpinski proved that ... 1answer 285 views ### Sum of Reciprocals of Primes in Imaginary Quadratic Field Diverges (2014 Miklós Schweitzer) Problem 5 of the 2014 Miklós Schweitzer states: Let \alpha be a non-real algebraic integer of degree two, and let P be the set of irreducible elements of the ring \mathbb{Z}[\alpha]. Prove that ... 3answers 3k views ### Non-increasing sequence of positive real numbers with prime index If a_n is a sequence of non-increasing positive numbers, then suppose we already know that$$\sum_p a_p$$converges, when p runs over the primes, what should be used to prove that$$\sum_n \frac{...
I'm doing a little project on the $\zeta$ function, and I am at a complete loss of what it is actually doing. I understand it is way over my head, but when I am plugging say $\zeta(1 + i)$ into ...