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On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.

2 votes
0 answers
114 views

Distribution of residue classes of totients of (univariate) polynomials

Let $\phi$ denote Euler's totient function and $f$ a non-constant univariate polynomial with integer coefficients such that $f(u) \in \mathbf N^+$ for all $u \in \mathbf N^+$ (assume $f$ is irreducibl …
Salvo Tringali's user avatar
5 votes
Accepted

How composite $a^n+b$ is?

This is an answer to the part of the question where the OP is asking if the image of the function $$R_{2,\varphi}: \mathbf N^+ \to \mathbf R: n \mapsto \frac{\varphi(2^n - 1)}{2^n-1}$$ is dense in th …
Salvo Tringali's user avatar
7 votes
3 answers
2k views

If the natural density (relative to the primes) exists, then the Dirichlet density also exis...

On p. 76 of the 1996 edition of Serre's A Course in Arithmetic, one reads the following (inline) remark: One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and …
Salvo Tringali's user avatar
2 votes

Reference to a variant of Abel's summation formula

Sorry for answering my own question, but after hearing from Zaccagnini I found a reference to the variant of Abel's summation formula mentioned in the early version of the OP, where $f^\prime$ was ass …
Salvo Tringali's user avatar
5 votes
1 answer
200 views

Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $...

Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) serie …
Salvo Tringali's user avatar
8 votes

$\zeta^{(k)}(s) < 0$ for $s\in (0,1)$

Here is a proof that $\zeta^{(i)}(s) < 0$ for all $s \in [0,1[$ and $i \in \mathbf N$ (I can't say whether it counts as brief and clean, though). We'll use that the Riemann zeta can be expanded as …
Salvo Tringali's user avatar
4 votes
2 answers
666 views

Reference to a variant of Abel's summation formula

Edit. A stronger version of the formula is true (details follow). Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, $(\lambda_n)_{n \ge 1}$ a nondecreasing sequence of positive reals such that …
Salvo Tringali's user avatar