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Results tagged with analytic-number-theory
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user 151669
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.
5
votes
More accurate formula for the Chebyshev $\psi$ function
The original result is due to Goldston "On a result of Littlewood concerning prime numbers. II".
This result has recently been made explicit, see (https://arxiv.org/abs/2107.14468v1).
Ramaré also has …
- 465
5
votes
Accepted
Conjectured error term when counting square-free integers
Your guess is correct! It is indeed conjectured that $a=1/4$. A good recent reference is [1]. In particular, it is known that
$$E(x)=\Omega(x^{1/4})$$
and computations have shown
$$|E(x)|<1.12543x^{1/ …
- 465
4
votes
The twin prime problem and the Jurkat-Richert Theorem
As discussed in Stanley's answer, the main overarching reason why we cannot prove the Twin Prime conjecture (or Goldbach's conjecture) is the parity problem. However, in terms of the specific lower bo …
- 465
4
votes
Accepted
Weights in the proof of Chen's theorem in Nathanson's "Additive Number Theory The Classical ...
Choosing optimal weights in sieve theory is a very difficult problem that is often done by trial and error. In Nathanson's book it seems as if he was attempting to produce the simplest and shortest ve …
- 465
3
votes
A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis
I wanted to give some additional remarks to Will Sawin's answer and the associated comments.
Carneiro, Milinovich, and Soundararajan certainly have the best result in literature (as far as I'm aware). …
- 465
3
votes
Bounds of zeta function near $\Re(s)=1$
In addition to GH from MO's answer, if one wishes to keep the $(\log|\Im(s)|)^{2/3}$ factor, then there is a very recent improvement due to Bellotti [1]. In particular, Bellotti proved that
$$
\zeta(s …
- 465
2
votes
Riemann–Von Mangoldt formula
In addition to 2734364041's answer, this paper of Tim Trudgian may be useful: in particular, Trudgian shows that for all $T\geq e$,
$$\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)-\frac{7 …
- 465