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4
votes
1answer
144 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
93 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
101 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 ...
1
vote
1answer
149 views

Question on an arithmetic function with the sieve of Eratosthenes

I want to ask some question related with the sieve of Eratosthenes. The sieve of Eratosthenes: write it as $E_1(x) (=\pi(x)-\pi(\sqrt x)+1)$. Then we have an obvious result $$E_1(x)/x\ln^{-1}x = ...
4
votes
1answer
154 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 ...
0
votes
0answers
100 views

asymptotics of primes in arithmetic progressions

If $a$ and $q$ are given coprime positive integers, what is the best known error term for $$ \sum_{p<x,\,p\,\text{is prime},\,p\equiv a \pmod q} \frac{\log p}p-\frac{\log x}{\varphi(q)}? $$ Is it, ...
6
votes
0answers
266 views

implication of divergence of $1/\zeta(s) $ at 1/2

$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function. This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$. Its convergence is unknown if $1/2< ...
0
votes
0answers
133 views

Similarities between counting primes and counting integers

Primes can be counted via the explicit logarithmic counting function $\psi(x) \thicksim \pi(x) \log(x)$: $$\displaystyle \psi(x) = x - \log(2\pi) - \frac12 \log(1- \frac{1}{x^2}) - \sum_{\rho} ...
5
votes
3answers
611 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
170 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
491 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
218 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
218 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
317 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
264 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
900 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
132 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
364 views

Name of a conjecture on difference of prime numbers? [closed]

Hello Dear there is a conjecture for which I do not know how it is called. The conjecture is: Every even number can be always written as the difference between two prime numbers. Could you ...
1
vote
2answers
361 views

Can $\epsilon$ be a generating function? [closed]

I would like to know if I could do something like: $\epsilon (0)\text{:=}0$ $\epsilon (n)\text{:=}\frac{4}{2 n-(-1)^n+1}$ and use it instead of a constant. As $n\rightarrow \infty$, $\epsilon ...
0
votes
1answer
385 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
806 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, ...
3
votes
1answer
1k views

Error term of the Prime Number Theorem and the Riemann Hypothesis

I have read that the Riemann Hypothesis is equivalent to $\pi(x)=\text{Li}(x)+O(\sqrt{x}\log x)$ Is there an analogous statement saying the Riemann Hypothesis is equivalent to $\pi(x)=\frac{x}{\log ...
12
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} ...
11
votes
5answers
2k views

Upper bounds for the sum of primes <= n

Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the sum of the first $n$ ...