# Questions tagged [analytic-number-theory]

A beautiful blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.

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### Is there a "convolution" of asymptotic growth?

Suppose that I have two asymptotic counts given by $$\#\{x \in [0,H] \cap \mathbb Z: f(x) \leq H\} \sim F(H)$$ and also $$\#\{x \in [0,H] \cap \mathbb Z: g(x) \leq H\} \sim G(H).$$ From these two ...
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### On Carmichael function and aliquot parts of odd perfect numbers

I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
1 vote
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### Partial exponential sums over lattice points of lattice cones

Consider the usual lattice $M:=\mathbb{Z}^2\subseteq\mathbb{R}^2$, and let $v_1,v_2\in\mathbb{Z}^2\subseteq\mathbb{R}^2$ be two non-zero lattice points which are $\mathbb{Z}$-linearly independent. ...
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### Curious infinite product, convergence, connection to prime numbers

I have been playing with the following function: $$f(x)=\frac{\pi x (1-x^2)}{\sin\pi x}\prod_{k=2}^\infty \frac{\sin(\pi x/k)}{\pi x/k}$$ It is hard to get correct numerical values. I'll start with ...
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### Research of average number of equivalence classes of solutions to generalised Pell's equation

Statement of the problem Firstly, consider the infamous Pell's equation: $x^{2}-dy^{2}=1$. Here $x$ and $y$ are integers and $d$ is a nonsquare integer. It is known () that all solutions of this ...
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### An 'onion-structure' for roots of a series associated to prime numbers?

The series $$\sum_{n=1}^\infty\frac{z^{p_n-n}}{n!}$$ associated to the sequence $p_1=2,p_2=3,p_3=5,p_4=7,p_5=11,\ldots$ of prime numbers defines a holomorphic function in the open disc of radius $e$. ...
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### Absolute error in linearly approximating the sum of sum-of-divisors function

The sum-of-divisors function is defined as $\sigma(k):=\sum_{\ell\mid k}\ell.$ It is well-known that $$\sum_{k\le x}\sigma(k)=\frac{\pi^2}{12}x^2+O(x\log x),$$ and therefore it seems natural to study ...
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### Divisor of given order in short intervals

Is the following Open question or Conjecture already known, or eventually settled ? Open question : For sufficiently large $x$ there is at least a positive integer in the interval $[x,x+\log^2(x)]$ ...
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### Possible regularisation for sum of function of primes

Consider the following sum of function of primes: $$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$ Here $p$ runs through all primes and $e$ is Euler's constant. We can see that the sum ...
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### Asymptotic bound of some number theoretic function

I asked this in stack exchange but did not get anything so I am posting it here. I am self-studying asymptotic behavior of some number theoretic function and the following question comes up. Let $n$ ...
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### A bias for runs in Legendre symbols?

$\newcommand\Legendre{\genfrac(){}{}{#1}{#2}}$An odd prime $p$ defines the sequence $\Legendre1 p,\Legendre2 p,\dotsc,\Legendre{p-1}p$ of values of the Legendre symbol describing the quadratic ...
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### Proving Mertens' theorem using the prime number theorem

Mertens' Theorem states that $$\sum_{p \leq x}\frac{1}{p} = \log \log x + M + O(1/\log x).$$ This is weaker than the prime number theorem; in fact according to the Wikipedia page, the prime number ...
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### Exact value of $0<\wp^{-1}(2^{-{2/3}})<2$ with $\wp$ the Weierstrass elliptic function

I am investigating solutions to the differential equation $$\ddot{y}(t)=6y(t)^2,\dot{y}(0), y(0)=y_0>0.\tag{1}\label{1}$$ Let $\wp(t)$ be the Weierstrass elliptic function with elliptic invariants ...
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### Cardinality of $\{ n_i + i^k: i \in \mathbb{N} \} \cap [1,T]$ where $\{n_i \}$ is all natural numbers in some order

Let $n_1, n_2, ...$ be a sequence of natural numbers such that $\{n_i: i \in \mathbb{N}\}$ as a set is all of natural numbers. Let $k$ be a positive integer. Is is possible to obtain a lower bound of ...
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### Is the set of all solutions $x > 0$ to $\pi(x) = \operatorname{li}(x)$ unbounded?

Is the set of all solutions $x > 0$ to the equation $\pi(x) = \operatorname{li}(x)$ unbounded? Is $\liminf_{x \to \infty} |\pi(x)-\operatorname{li}(x)|$ equal to $0$? Here, $\pi(x)$ denotes the ...
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### An identity among values of the logarithmic derivative of $\zeta(s)$

From some known special values of the Riemann zeta function and its derivative, one can show that $$\gamma =1+ \frac{\zeta'(2)}{\zeta(2)} -\frac{\zeta'(0)}{\zeta(0)}+ \frac{\zeta'(-1)}{\zeta(-1)}.$$ ...
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### Error or gap in "Modular Functions and Dirichlet Series", by Apostol

My question concerns Apostol's Chapter 7, Kronecker's Theorem with Applications. It's Theorem 7.11, page 156. I’m attaching the proof in question. There is a lot going on, but I’ve highlighted the ...
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### Second moment of $S(T)$ for Dirichlet L-functions

Let $S(T)$ denote the argument of the Riemann zeta function. Selberg established that $$\int_0^T |S(t)|^2 \text{d}t\sim\frac{T}{2\pi^2}\log \log T.$$ Let now $\chi$ be a Dirichlet character modulo $q$,...
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### Why is $\sum_{m=1}^{n}\frac{(\nu(m)-\log\log n)^2}{n\log\log n}=\int_{-\infty}^{\infty}\omega^2\, \mathrm{d}\sigma_n(\omega)$?

I was reading an article on Probabilistic Number Theory by M.Kac where I am not able to understand why a particular equation mentioned here in page $657$ equation $(7.7)$ is true? I do understand that ...
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### L_infinity norm of signed sums of Fourier characters and discrepancy of Fourier matrices

Consider signed sums $\displaystyle A_f(x) =\sum_{\chi} (-1)^{f(\chi)} \chi(x)$ for some set $S$ of characters of an abelian group $G$, and signing $f$ of the characters. For a fixed set $S$ what is ...
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### Average value of the prime omega function $\Omega$ on predecessors of prime powers

For a positive integer $n$, the prime omega function value $\Omega(n):=\sum_{p\mid n}{\nu_p(n)}$ counts the number of prime divisors of $n$ with multiplicities. A result of Hardy and Wright, [1, ...
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### The nontrivial zeros of the zeta function and the prime counting function

The truncated explicit formula has the shape \begin{equation}\label{1} \psi(x) =x-\frac{\zeta^{\prime}(0)}{\zeta(0)}-\sum_{|\rho|\leq T}\frac{x^{\rho}}{\rho}+\sum_{n=1}^{\infty}\frac{x^{-2n}}{2n}+...
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