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 tagged [analytic-number-theory]
3532 questions
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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 $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$
The Riemann hypothesis asserts that all the…
Karna
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Books about the Riemann Hypothesis
I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function.
The following are excluded:
Books by mathematical cranks (especially books by amateurs who claim to…
Marko Amnell
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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 what types of techniques are used and just how…
Sean Tilson
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Riemann zeta function at odd positive integers
Starting with the famous Basel problem, Euler evaluated the Riemann zeta function for all even positive integers and the result is a compact expression involving Bernoulli numbers. However, the evaluation of the zeta function at odd positive…
Dinesh
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$\forall n\in\mathbb N:n^x\in\mathbb Q$ implies $x\in\mathbb Z$ - elementary proof?
Consider the following two problems:
Show that if for some $x\in\mathbb R$ and for each $n\in\mathbb N$ we have $n^x\in\mathbb N$, then $x\in\mathbb N$.
Show that if for some $x\in\mathbb R$ and for each $n\in\mathbb N$ we have $n^x\in\mathbb Q$,…
Wojowu
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45
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What is the Todd's function in Atiyah's paper?
In terms of purported proof of Atiyah's Riemann Hypothesis, my question is what is the Todd function that seems to be very important in the proof of Riemann's Hypothesis?
Jose Garcia
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44
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Small primes attract large primes
$$
\begin{align}
1100 & = 2\times2\times5\times5\times11 \\
1101 & =3\times 367 \\
1102 & =2\times19\times29 \\
1103 & =1103 \\
1104 & = 2\times2\times2\times2\times 3\times23 \\
1105 & = 5\times13\times17 \\
1106 & =…
Michael Hardy
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How does $ \sum_{p
Note the $ p < x $ in the sum stands for all primes less than $ x $. I know that for $ s=1 $,
$$ \sum_{p 1 $, the partial sums actually converge to a finite limit called the prime zeta…
anon
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Are there infinite many $n\in\mathbb N$ such that $\pi(n)=\sum_{p\leq\sqrt n}p$?
Are there infinite many $n\in\mathbb N$ such that $$\pi(n)=\sum_{p\leq\sqrt n}p,\tag{1}$$
where $\pi(n)$ is the Prime-counting_function?
For example, $n=1,4,11,12,29,30,59,60,179,180,389,390,391,392,\dots$
As I know, $\pi(x)\sim \dfrac{x}{\ln…
lsr314
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Are these zeros equal to the imaginary parts of the Riemann zeta zeros?
Edit 8.8.2013: See this question also.
The Fourier cosine transform of an exponential sawtooth wave times $e^{-x/2}$:
$$\operatorname{FourierCosineTransform}(\operatorname{SawtoothWave}(e^x)\cdot e^{-\frac{x}{2}})$$
can be plotted with the following…
Mats Granvik
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36
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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 sieves, but I became intrigued when I read over the…
davidlowryduda
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34
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Sequence of numbers with prime factorization $pq^2$
I've been considering the sequence of natural numbers with prime factorization $pq^2$, $p\neq q$; it begins 12, 18, 20, 28, 44, 45, ... and is A054753 in OEIS. I have two questions:
What is the longest run (a subsequence that consists of…
David Bevan
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Elementary proof that the limit of $\sum_{i=1}^{\infty} \frac{1}{\operatorname{lcm}(1,2,...,i)}$ is irrational
Show that the infinite sum $S$ defined by -$$S=\sum_{i=1}^\infty \frac{1}{\operatorname{lcm}(1,2,...,i)}$$ is an irrational number.
I found this question while reading 'Mathematical Gems' by Ross Honsberger. After pondering over it for nearly an…
TheBigOne
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Sums of the form $\sum_{d|n} x^d$
Let $$S(x,n) = \sum_{d|n} x^d, \quad n \in \Bbb N. $$
Do these sums appear in the literature? What are they called if they do and what is known about them?
To clarify, note that this sum is not the same as the generalized divisor function
$$…
Asvin
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32
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Rounding is asymptotically useless?
Recently I came across the nice result that
$$\left\lfloor n \right\rfloor - \left\lfloor\frac{n}{2}\right\rfloor + \left\lfloor\frac{n}{3}\right\rfloor - \left\lfloor \frac{n}{4}\right\rfloor + \dots \sim n \log 2$$
where $\displaystyle a_n \sim…
Aryabhata
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