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### A function asymptotical equivalent with the prime counting function?

This is closely connected to Schinzel's Hypothesis H and the Bateman-Horn conjecture. A slightly different way to phrase this question would be to consider the two polynomials $(x, x^2 + 4)$, and look ...

### Tate gamma factor as a principal value integral

For the second question, let $\varpi$ be a uniformizer, and set $\mathfrak p = \varpi \mathcal O_F$. Let's define the conductor of $\chi$ to be $f$ if $\chi$ is trivial on $1+\mathfrak p^f$, but not ...

### Change of fractional numbers from one base to another

I'll assume you know how to write a fraction $A/B$ in decimal form, i.e. $A/B=(0.d_1d_2...)_{10}$. Then, given that $x=(0.a_1a_2...)_b=\sum_{i=1}^\infty a_i/b^i,$ just write each successive term in ...
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### How to find all natural numbers $n$ such that $a \mid n$ implies $(a+1) \mid (n+1)$?

Let $n$ be a good number. Claim 1. $n$ is not a composite number. Claim 2. $1$ is a good number (which is obvious since $(1+1) \mid 2$). Claim 3. All primes but $2$ are good. In conclusion, we've ...