The prime-number-theorem tag has no usage guidance.

**-1**

votes

**1**answer

175 views

### Number of different factors of given size in primorial

Let $b_n$ be number of bits in product of all primes from $1$ to $n$ which is approximtely $b_n\approx n$.
What is the approximate number of distinct factors with number of bits ...

**7**

votes

**0**answers

336 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 ...

**6**

votes

**0**answers

299 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< ...

**3**

votes

**0**answers

129 views

### Does data suggest $| \pi_2 (n) - 2\Pi \int_2^n \frac{dx}{\ln(x)^2} | < \ln(n+2)^2 \sqrt (n+2) $?

Let $\Pi$ be the twin prime constant and $\pi_2(n)$ the twin prime counting function.
Define
$$ t(n) = \left| \pi_2(n) - 2 \Pi \int_2^n \frac{dx}{\ln(x)^2} \right| $$
Is it consistent with current ...

**1**

vote

**0**answers

136 views

### Siegel Walfisz Theorem for algebraic number fields

Is there a generalization of the Siegel Walfisz to algebraic number fields? This has been done for the prime number theorem in the prime ideal theorem.

**1**

vote

**0**answers

154 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 ...

**1**

vote

**0**answers

153 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

**0**answers

113 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, ...