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

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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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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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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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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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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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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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$$
\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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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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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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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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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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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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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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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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