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Results tagged with analytic-number-theory
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user 10811
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.
24
votes
Function zeros in strip 0 < Re < 1
The question for the case of a linear combination of Dirichlet L-series is
actually easier than the case of a single L-function (Since RH is not known).
In fact in each strip $1/2 \leq \sigma_1 <\Re …
13
votes
Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?
Pintz paper from arxiv last week
"Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture" http://arxiv.org/abs/1305.6289,
p …
- 2,913
22
votes
4
answers
6k
views
How does Yitang Zhang use Cauchy's inequality and Theorem 2 to obtain the error term comin...
I have been reading Yitang Zhang's paper now for one and a half weeks and also volunteered to give a popular talk on the paper next week at Stockholm University.
Today I found a detail in the proof …
- 2,913
10
votes
What are the analytic properties of Dirichlet Euler products restricted to arithmetic progre...
Well, Peter's answer is overkill for this particular problem. While this zeta-function will certainly be a Burgess zeta-function, the study of the zeta-function of this particular kind will be much si …
- 2,913
20
votes
Accepted
Density of prime pairs whose gap is less than the average gap
It is conjectured (see e.g. Goldston-Pintz-Yildirim http://arxiv.org/abs/1103.5886) that
$$
\lim_{n \to \infty}\frac{1}{n}\#\Big\{m \le n : \alpha<\frac{p_{m+1} - p_m}{\ln p_m} < \beta \Big\} = \i …
- 2,913
10
votes
Accepted
lower bound for $\Re\zeta(1+it)$
There is no lower bound for Re$(\zeta(1+it))$. This can be found in Lamzouri's paper http://www.math.uiuc.edu/~lamzouri/distribzeta.pdf (see e.g. page 3, formula (5)). (Choose $\tau$ large enough, $\ …
- 2,913
6
votes
non-trivial zeros of partial zeta functions
Micah's answer answers your Q1. My reply gives some information on your Q2. Let us assume that $1 \leq a\leq N$ for simplicity (this condition can be removed). Your zeta-functions $\zeta$ and $\Psi$ …
- 2,913
5
votes
Accepted
Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$
I believe the comments of joro and Carlo Benakker right under your question is to the point. Since the zeroes close to $s$ will be the ones that contributes in the sum, the zeroes close to s must be c …
- 2,913