The prime-number-theorem tag has no wiki summary.

**0**

votes

**0**answers

76 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

**1**answer

133 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

**0**answers

79 views

### Where to include contact details in math paper? [closed]

I recently submitted a paper to a math journal on a prime number patter using latex formatting, but they sent an email back saying that the contact details for the corresponding author should be in ...

**4**

votes

**1**answer

143 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

**0**answers

97 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

**0**answers

250 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

**0**answers

126 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} ...

**6**

votes

**3**answers

594 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

**0**answers

169 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 ...

**0**

votes

**0**answers

178 views

### Period of decimal for $1/n$, odd part of $n+1$, and primes.

Let $n$ be a nature number is relatively prime to 10,such the period of the decimal expansion of $1/n$ is $n-1$ or a divisor of $n-1$, and let $c$ be the "cycle length of $n$" (defined below). If ...

**7**

votes

**3**answers

458 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

**1**answer

199 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

**1**answer

199 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

**4**answers

312 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

**0**answers

244 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 ...

**21**

votes

**2**answers

2k 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

**2**answers

867 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

**0**answers

128 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

**1**answer

359 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

**2**answers

358 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

**1**answer

382 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

**3**answers

795 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

**1**answer

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 ...

**10**

votes

**3**answers

1k 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

**5**answers

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 $n$ ...