# Questions tagged [analytic-number-theory]

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

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### Asymptotic density of numbers of the form $p_{1}^{\alpha_{1}^{2}}\cdot p_{2}^{\alpha_{2}^{2}}\cdots p_{k}^{\alpha_{k}^{2}}$

If $n$ is a number of the form $p_{1}^{\alpha_{1}^{2}}\cdot p_{2}^{\alpha_{2}^{2}}\cdots p_{k}^{\alpha_{k}^{2}}$ (OEIS A197680) and $T(x)$ counts how many of these numbers are between $1$ and $x$, ...
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### Probabilistic proof of existence of an integer

The prime number theorem (PNT) says that an integer $n$ is prime with probability $\frac{1}{\ln n}$. Using only PNT, it's conceivable that each integer upto $10^{10^{10}}$ is non-prime. However using ...
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### A simple property of Kloosterman sum

Kloosterman sum is defined as $$K(a,b;m)=\sum_{0\leq x \leq m-1}_{\gcd(x,m)=1} e^{2\pi \mathcal{i} (ax+bx^*)/m}$$ where $a,b,m \in \mathbb{N}$ and $x^*$ is the inverse of $x$ modulo $m$. Now there ...
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### Express Dirichlet series in terms of Dirichlet L-function

Assume $\gcd(a,b)=1$, how to express $$F(s)=\sum_{n \equiv a \pmod b} \frac{\mu(n)}{n^s}$$ in the half plane where $\Re(s)>1$ in terms of Dirichlet L-function?
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### Will partial summation work for this problem?

Define $$G(x)=\sum_{n \leq x} T\left(\frac{x}{n}\right)$$ and $G,T: [1,\infty) \to \mathbb R$ And function T satisfies the following conditions: 1) $T(x)=O(x)$ 2) $T(x) \sim cx (x \to \infty)$ How ...
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### Quadratic characters and Liouville's function

I'm working through the problems in Montgomery & Vaughan's Multiplicative Number Theory. In Section 11.2 'Exceptional Zeros', Exercise 9a says that for a quadratic character $\chi$, show that for ...
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### Prime asymptotics from Euler product

It is said that the Euler product $$\prod_p \frac{1}{1-p^{-s}}$$ diverges as $s \to 1^+$ proves we can't find constants $C$,$\theta$ with $\theta < 1$ such that $\pi(x) < C x^\theta$ because ...
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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 ...
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### Periodic Zeta Function Functional Equation

Recall that the periodic zeta function has the Dirichlet series $$F(\lambda,s)= \sum_{n=1}^\infty \frac{e^{2\pi i n\lambda}}{n^s}.$$ This defines an analytic function for $\Re s>0$ and has a ...
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### On functions similar to Hurwitz zeta function

Denoted as $\zeta(s,a)$ for a > 0 Where do I find topics on the Hurwitz zeta function for a < 0? Any links or resources would be appreciated. (Please dont mention wiki or mathworld) Thanks
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### Equidistribution results vs transcendence degree

Consider $\alpha =(\alpha_1, \dots, \alpha_n) \in \mathbb{R}^n$ linearly independent over $\mathbb{Q}$, then the map $q \mapsto q \alpha:= ( q \alpha_1, \dots, q \alpha_n)$ gives a dense embedding ...
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### Difference between zeta sum and Euler product?

The fact that $$\sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p}\frac{1}{1-p^{-s}}$$ is a consequence of unique factorization of primes. We could form a similar sum and a similar product of irreducibles ...
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### Asymptotic formula for d(n)/n summation

I was trying to show $$\sum_{n \le x} \frac{d(n)}{n} = \frac{1}{2}\log(x)^2 + 2\gamma \log(x) + O(1)$$ where $d(n)$ is the number of divisors of $n$ and $\gamma$ is the Euler constant using the ...
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### Supplemental number theory text to Montgomery and Vaughan

We already have a large list of the Best book ever on Number Theory, but I'm looking for a more targeted response for analytic number theory. Specifically, I'm taking a trip on which I may or may ...
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### Question on de la Vallee Poussin's simplified proof of Dirichlet's theorem on primes in arithmetic progressions

I've been trying to understand de la Vallee Poussin's "Demonstration Simplifiee du Theorem de Dirichlet sur la Progression Arithmetique" and I've got stuck at the following step on pg 18 where Poussin ...
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### Proving $\pi(\sqrt{p_{1}p_{2}\cdots p_{n}})>2n$ for $n \geq 6$

I am having trouble in solving the following problem. Let $p_{n}$ denote the $n$-th prime. Then prove that $$\pi(\sqrt{p_{1}p_{2}\cdots p_{n}})>2n$$ for $n \geq 6$. No idea how to start.
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### Why are complex numbers necessary to prove the Prime Number Theorem?

The standard proof of the Prime Number Theorem requires taking into consideration that there are no zeroes of the Riemann Zeta function in which the real part equals one. But consider the following ...
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### Bounding the series $\sum_{m\leq x,m\neq n}\frac{1}{|\log(m/n)|}$

I am trying to reproduce the following bound: $\sum_{1\leq m\leq x, m\neq n}\frac{1}{|\log(m/n)|}=O(x\log(x))$, for $x\geq 2$ and some $n$, $1\leq n\leq x$ (the implicit constant shouldn't depend on ...
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### Expressing $\pi(x) = \frac{P(x)}{Q(x)}$, where $P$ and $Q$ are polynomials

In Tom Apostol's Analytic Number Theory book there is a problem which states: That there do not exists polynomials $P(x)$ and $Q(x)$ such that $$\pi(x) = \frac{P(x)}{Q(x)}$$ for all $x \in \mathbb{N}$...
In Titchmarsh's book "The theory of the Riemann Zeta function" pg. 15 where the functional equation of the zeta function is being derived, I couldn't understand this part: $$\frac{s}{\pi} \sum_{n=1}^{... 2answers 1k views ### Infinite product representation of a function in terms of its non-trivial zeroes? From Wikipedia's Weierstrass Factorization Theorem, I learned that every entire function can be represented as a product involving its zeroes. Examples are the sine and cosine function. The Riemann ... 1answer 160 views ### Nonnegativity of the quadratic Dirichlet L-function L(\tfrac{1}{2},\chi) under GRH I have been looking for a proof of the statement: "Assume the Generalized Riemann Hypothesis. Let d be a fundamental discriminant and \chi_d the associated primitive quadratic character. Then,$$L(...
For Riemann zeta function and $L$-functions of number fields, the width of critical strip is $1$. For $L$-functions of modular forms of weight $k$, the width of the critical strip is $k$. Why is ...