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Results tagged with nt.number-theory
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user 133811
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
3
votes
Accepted
A simple question about the classical divisor problems
It is known (see Titchmarsh Chapter 12) that if you define $\gamma_k$ the lower bound of $\sigma > 0, \int_{-\infty}^{\infty}\frac{|\zeta(\sigma+it)|^{2k}}{|\sigma+it|^2}dt < \infty$, then $\frac{k-1} …
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1
vote
Upper bound for a subset of $\mathbb{N}^2$
Case 1: $m>0$, then $m \leq 4N^2$, so $d(m)=O((4N^2)^{\epsilon})= O(N^\epsilon)$, but then $c=a-b, d=a+b$ are divisors of $m$ so the number of such pairs lies in a $O(N^\epsilon)^2=O(N^\epsilon)$ set, …
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3
votes
On the spacing of the zeros of the Riemann zeta function
On the critical line, zeros are analyzed with the Hardy function which is real and has very good approximation formulas (Riemann-Siegel)- tons of references online including a fairly good book by Ivic …
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6
votes
Can any Hurwitz zeta function be written as an Euler product?
To add a little to the excellent answer above - it is known that for $0<a<1, a \ne 1/2$ the Riemann Hurwitz function has a lot of zeroes in the strip $1 < \sigma < 1+a$, so, in particular, there canno …
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3
votes
On Exercise 2.5.10 in Ram. M. Murty's book, "Problems in Analytic Number Theory."
The computation of the powers is wrong and the result stated in the book is incorrect and it should be $cx+ O(x^{(1+\delta-\delta^2)/(2-\delta)})+O(x^{\frac{(1-\delta)(1+\alpha)}{2-\delta}})$
If $y=x …
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2
votes
The Dirichlet series of the harmonic numbers
Let me put my comment in an answer since it seems to solve the problem at least in the analytic continuation, poles, residues etc way.
Fix a level $k \ge 1$, then using the asymptotics $H_n=\log n+\ga …
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6
votes
Is there a collection of evidence and heuristic arguments against the Riemann hypothesis?
(note that the original question before being edited out had an argument about why RH is false and the post below was a refutation of that); the Ivic paper linked by @Mayank contains some good argumen …
6
votes
Where can I find the problem by Lagarias?
The easiest way is to use Ivic's inequality
$\sigma(n)<2.59n \log \log n, n \ge 7$
Then $H_n>\log n +\gamma$ so $e^{H_N}>e^{\gamma}n$ and $\log H_n > \log \log n$ for $n \ge 3$ hence:
$2\exp(H_n)\log( …
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6
votes
Does $\int_{2}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$ converge on the real axis for $s...
the answer to the original question is obviously we do not know (as noted the question is equivalent to RH), while the reasoning above doesn't work for the same reason that, for example, the fact that …
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3
votes
A generalization of strong primes
As requested I put my comment as a (partial) answer:
Since we know there are infinitely many consecutive primes with finite gap $C$ (smallest proved $C$ as of now seems to be $246$), any $\theta$ for …
- 1,795
12
votes
Accepted
A question on an equivalence of RH
Note that if $1/2< \sigma <1, t \in \mathbb R$ one has $\frac{2\sigma-1}{\sigma^2+t^2} < \frac{2\sigma-1}{(1-\sigma)^2+t^2}$.
By a little manipulation, one gets:
$\frac{2\sigma}{\sigma^2+t^2} + \frac{ …
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27
votes
Why is so much work done on numerical verification of the Riemann Hypothesis?
I would add a few more comments to the very pertinent ones above:
1: We are lucky to have two things that work in our favor - an excellent representation of $\zeta$ on the critical line by a simple …
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2
votes
Is there an analogue of the Balazard-Saias-Yor criterion for the Riemann Hypothesis for fini...
There is no Weil Zeta Function per se, but a bunch of such associated with various algebraic-geometric objects (the simplest ones are associated with elliptic curve structures on toruses); such functi …
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4
votes
Accepted
Fourier coefficients of Selberg polynomials
As it is odd, we can write the Vaaler polynomial $V_K(x)=\sum_{1 \le k \le K}c_{k,K} \sin 2\pi kx$ and its fundamental property is that $|c_{k,K}| \le \frac{1}{\pi k}$.
This follows easily from its de …
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