3
votes
1answer
309 views

Lower bounds on the error term of the prime number theorem

Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t. $$f(x)\ll |\psi(x) - x|$$ where $\psi$ is the Chebyshev function.
6
votes
2answers
215 views

Divisor sums over values of binary forms of primes

Let $\tau$ be the divisor function, that is $$ \tau(n)=\sharp\{d \in \mathbb{N}, d|n\}. $$ I was wondering if anyone has ever proved an asymptotic estimate for the sum $$S(x):=\sum_{p,q\leq ...
9
votes
1answer
305 views

Squarefree numbers $n$ such that $432n+1$ is also squarefree

This is a second attempt (see Primes $p$ such that $432 p +1$ is prime) Is the set of squarefree numbers $n$ such that $n(432 n+1)$ is also squarefree known to be infinite? Fact: the number of such ...
-1
votes
1answer
296 views

Primes $p$ such that $432 p +1$ is prime [closed]

Is the set of prime numbers $p$ such that $432 p + 1$ is also prime infinite? It doesn't follow from Dirichlet's theorem as far as I can tell.
5
votes
2answers
572 views

The shortest interval for which the prime number theorem holds [closed]

It is well known that the prime number theorem on the form \begin{align*} \pi(x+y) - \pi(x) \sim \frac{y}{\log (x+y)} \end{align*} breaks down for short enough intervals, e.g. taking $y=(\log ...
4
votes
1answer
153 views

Log weight removal in general (weaker) prime number theorem

Let $a_n$ be a sequence of non-negative numbers. Assume that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} a_p\log p}{X}\leq 1.$$ Can we prove that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} ...
1
vote
0answers
101 views

Prime counting function with a form of finite product using perron's formula

There's a form of complex integral what Riemann obtained to finding $\pi (x)$, $$ \pi^{*}(x)=\int_{L}\frac{\log \zeta (s)}{s}x^{s}ds, (1)$$ we already know that it lead us to the Prime Number ...
0
votes
0answers
110 views

Does Riemann's explicit formula imply invariance of the prime gaps distribution under a Fourier-like transform?

Loosely speaking, Riemann's explicit formula states that there exists a Fourier-type duality between the primes and the non trivial zeroes of the Riemann zeta function. Does this mean that the ...
4
votes
1answer
157 views

Consecutive Primes mod 3

Is anything known asymptotically about the binary "primes mod 3" sequence besides Dirichlet's result that 1 and 2 occur half of the time? For example, can you prove that it does not eventually cycle ...
5
votes
3answers
629 views

a question for the prime counting function

A famous inequality that has been proved by J.B. Rosser and L. Schoenfeld says that $\frac{n}{\ln n-1/2}$ < $\pi(n)$<$\frac{n}{\ln n-3/2} , n\ge 67$. Using this inequality we can prove ...
1
vote
0answers
171 views

apparent contradiction from result on prime gaps [closed]

I'm looking at Theorem 3 of this paper, which is If $\:x\geq \exp(\exp(45))\:$ and $\;\;h\:\geq\:3\cdot x^{\frac23}\;\;$ then $$\pi(x+h)-\pi(x) \; \geq \; h\cdot \left(1-\left(3192.34\cdot ...
8
votes
3answers
508 views

Asymptotic bounds on $\pi^{-1}(x)$ (inverse prime counting function)

What are the current best asymptotic bounds on $\pi^{-1}(x)$, where $\pi(x)$ denotes the prime counting function (number of primes at most $x$)? In other words, I am curious about the state of the ...
2
votes
1answer
221 views

Given an even integer N, what is the minimum set of primes such that any even number x <= N can be expressed as the sum of two primes from the set?

Given an even integer N, what is the minimum set of primes such that any even number $x \leq N$ can be expressed as the sum of two primes in the set? Goldbach's conjecture said Every even integer ...
4
votes
1answer
229 views

Estimate on the prime-counting function $\psi(x)$.

There is an elementary statement that I believe I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard) and if so, where I ...
0
votes
4answers
318 views

The prime number $2$ [duplicate]

Possible Duplicate: Why is 2 so odd? I have read few books and articles, almost all of them refer that any prime $p>2$. Just wondering why it has to be $>2$?
7
votes
0answers
272 views

Montgomery's conjecture and lower bound on certain Fourier transform.

Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...
22
votes
2answers
3k views

Why do primes dislike dividing the sum of all the preceding primes?

I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...
6
votes
2answers
906 views

Probability that randomly chosen integers from a restricted set of natural numbers are coprime

We know that the probability $P(k)$ of $k$ randomly chosen integers $(k \ge 2)$ from the set of natural number are coprime is $$ P(k) = \frac{1}{\zeta(k)}. $$ I am looking at a special case of ...
1
vote
0answers
134 views

Limit of Sequence of unusual Prime Product

Let $p_n$ be the nth prime and $p_L$ be closest to its square root: \begin{equation} p_L^2 \approx p_n \approx x \end{equation} Let $\sigma \in Z^+$ be a positive integer constant. Define the ...
0
votes
1answer
386 views

Primes are to Irreducible Polynomials as Prime-related theorems are to ?? [closed]

Irreducible polynomials are often introduced as the analog to prime numbers in polynomial rings. Prime numbers, of course, have a very rich theory, leading to the likes of the Riemann Zeta function ...
7
votes
3answers
815 views

Asymptotics for primality of sum of three consecutive primes

We consider the sequence $R_n=p_n+p_{n+1}+p_{n+2}$, where $\{p_i\}$ is the prime number sequence, with $p_0=2$, $p_1=3$, $p_2=5$, etc.. The first few values of $R_n$ are: 10, 15, 23, 31, 41, 49, 59, ...
13
votes
3answers
2k views

Why is the Chebyshev function relevant to the Prime Number Theorem

Why is the Chebyshev function $\theta(x) = \sum_{p<=x}\log p$ useful in the proof of the prime number theorem. Does anyone have a conceptual argument to motivate why looking at $\sum_{p<=x} ...