All Questions
Tagged with analytic-number-theory prime-numbers
713
questions
1
vote
1
answer
139
views
Given the primes, how many numbers are there?
I have a concrete question and a more abstract version of the same. There are lots of theorems that take information on the number of elements and translate it to information on the primes; I’m ...
- 32.3k
1
vote
1
answer
66
views
Sums involving reciprocal of primes
I am interested in obtaining an upper bound for
$$
\sum_{j=1}^N 1/p_j
$$
and upper+ lower bound for
$$
\sum_{p|K} 1/p.
$$
Here $p_j$ is the j-th prime and $p$ is prime.
I was able to find an ...
- 2,913
0
votes
0
answers
39
views
Normal Order of Distinct Prime Factor $\omega(n)$
Define $\omega(n)$ as number of distinct prime factors $n$ has, that is if $n=p_1^{a_1}... p_k^{a_k}$, then $\omega(n)=k$.
It is commonly understood that normal order of $\omega(n)$ is $\log\log(n)$, ...
- 43
0
votes
1
answer
35
views
lower bound of $\sum_{n=1}^x \frac{\mu(n)}{n}$
Denote by $\mu$ the Mobius function. Poussin showed that
$$
\sum_{n=1}^x \frac{\mu(n)}{n} = O(1/\log x),
$$
and there are further improvements since. I wonder what is known about lower bound of ...
- 175
1
vote
1
answer
87
views
The sum $f(x)= \sum_{p \le x} e^{\frac{1}{p\log p}}$ is very close to $\pi(x)$. Why is that?
The prime counting function $\pi(x)$ counts the number of primes less than a given $x$. There are other counting functions like Chebyshev's functions which count sums of logarithms of primes up to a ...
- 754
4
votes
1
answer
192
views
Does this function in $3$b$1$b has a name?
I was watching this video from $3$Blue$1$Brown channel, at minute 21:00 he introduced the following function:
$$
\chi(n)=
\begin{cases}
0 & \text{if } n=2k \\
1 & \text{if } n=4k+1\\
-1 & \...
- 6,100
0
votes
1
answer
90
views
Understanding a key definition in Lagarias Odlyzko's paper on Chebotarev density theorem
Lagarias & Odlyzko has a 1977 paper where they prove effective versions of the Chebotarev density theorem. I am having trouble understanding equation (3.1).
Here, $L/K$ is a Galois extension of ...
- 787
1
vote
0
answers
39
views
Evaluation of sum of product of dirichlet characters
An Analytic number theory question:
Let, $\chi_{0,q_2}$ be the trivial character modulo $q_2$ and $rad(q)$ be the radical of $q$.
I need to estimate the summation below:
$$S_1=\sum\limits_{\substack{...
- 11
1
vote
0
answers
51
views
Bounding $\psi(x)-x$ outside a set of finite logarithmic measure
By a 1980 result of Gallagher, assuming the Riemann hypothesis, one has $\psi(x)-x = O(x^{1/2} (\log \log x)^2)$ outside a set of finite logarithmic measure. I'm wondering what the state-of-the-art ...
- 877
0
votes
1
answer
60
views
Lower bound for the prime zeta function
The prime zeta function is defined for $\mathfrak{R}(s)>1$ as
$P(s)=\sum_{p} \ p^{-s}$, where $p \in \mathbb{P}$.
It is well-know this series converges whenever $\mathfrak{R}(s)>1$.
Now, ...
- 85
3
votes
1
answer
95
views
Density of squares using large sieves
I am reading Serre's Lectures on the Mordell-Weil Theorem, where he specifically talks about a Large Sieve inequality and proceeds to give an example.
Theorem. (Section 12.1) Let $K$ be a number ...
- 600
1
vote
1
answer
61
views
a small doubt in the proof of the quantitative form of the prime number theorem
I have been studying the proof of the prime number theorem in the quantitative form as in Theorem 6.9 of Montgomery & Vaughan's book "Multiplicative Number Theory, which focuses on proving ...
- 11
1
vote
0
answers
45
views
How to correct the error between $\log(x!)\approx x\sum_{n\leq x}\delta(n)\frac{\log(n)}{n}$, where $\delta(n)$ is the density of primes near x?
Well assuming that the Prime Number Theorem is true, when substituting $\delta(x)$, the density of primes near $x$—which I am being vague of what it means 'cause I don't have enough foundation about ...
- 177
0
votes
0
answers
110
views
Number theoretic partial differential equation
Consider the equation
$$t\frac{\partial^2}{\partial t^2} \sum_{n=1}^\infty \Phi_n(x,t)=-x\frac{\partial}{\partial x}\sum_{n=1}^\infty \Phi_n(x,t)+\sum_{n=2}^{\infty}\Lambda(n)\Phi_n(x,t)$$
where $\...
- 754
4
votes
1
answer
297
views
Newman's Short Proof of Prime Number Theorem
I'm going through the paper of D. Zagier on Short Proof of Prime Number Theorem. There it says in V that
$\Phi(s)=\int_1^\infty \frac{d\vartheta(x)}{x^s}$ . Can someone please explain in details why ...
- 55