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Results tagged with analytic-number-theory
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user 106239
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.
1
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555
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On Robin's criterion for the Riemann Hypothesis
Statement 1 : (Robin) proved that if the R.H. is false then there exist constants $0<\beta <\frac{1}{2}$ and $c>0$ small , such that $\sum \limits_{d|n} d \geq e^\gamma n \ln \ln n+ n\frac{ c \ln \ln …
- 185
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Bound for $|p_n - \operatorname{li}^{-1}(n)|$
It is well-known that $|\pi(x)-\operatorname{li}(x)| \leq \epsilon(x)$, where $\pi(x) = \sum \limits_{p \leq x} 1$ is the prime counting function, where $\operatorname{li}(x) = \int \limits_{2}^{x}\fr …
- 185
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What is the upper bound for $\int \limits_{2}^{x} \frac{e^{-0.3\sqrt{\ln(t)}}}{\ln^2(t)} dt$?
For start, is $\int \limits_{2}^{x} \frac{e^{-0.3\sqrt{\ln(t)}}}{\ln^2(t)} dt \leq x\ln(\ln(x)) \frac{e^{-0.3\sqrt{\ln(x)}}}{\ln^2(x)}$ ?
If the above is true, what is a better bound for the integral …
- 185
7
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562
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Upper bound for $p_{n^2} - p_{(n-1)^2}$?
What is the best unconditional upper bound for $p_{n^2}-p_{(n-1)^2}$ such that $p_n$ is the $n$-th prime number?
Asymptotics suggest it's somewhere near $4 n \ln n$, but how to prove this?
Edit: it' …
- 185