# Tagged Questions

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### 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.
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### Smallest prime in an arithmetic progression

Let $\{a_n\}_{n\in\mathbb{N}}$ be defined as $a_n = a + bn$ for some $a, b >0,(a, b) = 1$. Are there good bounds on the minimal $k$ s.t. $a_k$ is prime. It is well known that there are infinitely ...
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### lower and upper bound for $\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k$?

Are there known any lower and upper bounds for $$\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k,$$ where $\Omega(n)$ is the number of prime factors counting multiplicities of $n$? Or at least is it known ...
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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 ... 1answer 308 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 ... 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. 2answers 577 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 ... 0answers 144 views ### Arguments for the second Hardyâ€“Littlewood conjecture being false? Assume that x,y > 2, and that x<y. Then the second Hardyâ€“Littlewood conjecture states that\pi(x + y) - \pi(y) \leq \pi(x).$$We can easily justify this heuristically, since$$ ...
Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be ...