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Given a fixed $a$ and $q$, this should be true for sufficiently large $n$ by the explicit versions of Dirichlet's theorem on arithmetic progressions. We cannot prove that this is true for every $n$. T …

The set $S$ is very likely finite. It is unclear if you intend for $a$, $b$, $c$ and $d$ to be positive. If you don't assume that $a$, $b$, $c$ and $d$ are positive, then $n!$ has such a representati …

The following question is inspired mostly by this question, answer and the comment by Wojowu there A naive approach to understanding odd perfect numbers is to make a Dirichlet series where the $n$th o …

Since we have good asymptotics for $\pi(n)$ by the prime number theorem (and can get good explicit bounds on that from Rosser and Schoenfeld's work as well as later work such as that by Dusart) this q …

One would have an ineffective but strengthened version of the Prime Number Theorem. A consequence of this would be there need to be some $\epsilon>0$ such that there's no zero in the strip with real …

Your conjecture for sufficiently large $n$ is implied by Cramer's conjecture. In general though, conjectures like this unless they are coming from some specific application aren't that interesting. It …

Not a complete answer, but a bit too long for a comment: Conjecture 1 is very likely to be very difficult if true. The corresponding conjecture for general primes is open. Let $p_n$ be the $n$th prime …

Sieve theory as such is generally considered to have started with Brun's 1915 and 1919 papers. The titles are "Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare" and ""La série $1/5+1/7+1/ …

There have been some good answers already given but I want to note another aspect, namely a heuristic involving the Möbius function. Let $\mu(n)$ be the Möbius function. The Riemann Hypothesis is equi …

In general, problems involving the composition of multiplicative functions are very hard to analyze. I don't see any specific way to approach this problem, and I'm skeptical that this is likely to be …

There's been a lot of work on unconditional results of this sort. Rosser and Schoenfeld showed in a 1962 paper that one can take $$\dfrac{e^{-\gamma}}{\log x} \left(1- \frac{1}{2\log^2 x} \right) < …

This should be provable by standard although laborious methods. What follows is a proof sketch (I have not checked all the computational details but this method should work). We recall a few basic fac …

Theorem: For any constant $c$ there are infinitely many primes $p_k$ such that there are at least $c$ primes between $p_k^2$ and $p_{k+1}^2$. Proof: Fix a $c$. Assume that for sufficiently large $k$ …

Conjecture 2 seems to be true. If $n \geq 2$ then $$\frac{1}{2}(k^2+k+2)k^{n-k-1}-1 \leq \left\{{n \atop k}\right\} \leq \frac{1}{2}{n \choose k} k^{n-k} < 2^n k^{n-k}. $$ (Inequalities from Here.) …

As far as I'm aware, we don't have any substantially non-trivial bounds on the behavior of $\psi(n)$ when $n$ is an odd perfect number. We can at least prove the following but none of these are diffi …