# Tagged Questions

**3**

votes

**1**answer

327 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

**2**answers

228 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 ...

**10**

votes

**1**answer

314 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

**1**answer

299 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

**2**answers

594 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

**1**answer

154 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

**0**answers

104 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 ...

**6**

votes

**0**answers

271 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< ...

**4**

votes

**1**answer

231 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 ...

**7**

votes

**0**answers

275 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 ...