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Results tagged with nt.number-theory
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user 110710
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
6
votes
Accepted
Some equations similar to Goormaghtigh problem
There are counter-examples to all three claims in the question above, so none of the claims are correct.
1- A counter example is {x,n,y,m,t}={2,5,5,3,31} as was indicated in a comment above.
Another …
- 1,001
1
vote
Why does not a zeta zero counting function $N_0(T)$ behave exactly in a neighborhood that al...
Mathematica does not perform the correct analytic extension of $\arg(\zeta(\frac12+i T))$ which I discovered through the related question at https://math.stackexchange.com/q/4037391 and the answer pos …
- 1,001
1
vote
An evaluation of the second Chebyshev function
For
$$\psi(x)=\sum\limits_{n=1}^x \Lambda(n)$$
and
$$T(x)=\sum\limits_{n=1}^x \log(n)$$
one has
$$\psi(x)=\sum\limits_{n=1}^x \mu(n)\, T\left(\frac{x}{n}\right)$$
and
$$T(x)=\sum _{n=1}^x \psi \left(\ …
- 1,001
8
votes
Accepted
Does this partial sum over primes spike at all zeta zeros?
This is not a complete answer but hopefully provides some insight.
The function $$\exp\left(-\sum\limits_{p\le x},\frac{\cos(x\, \log(p))}{\sqrt{p}}\right)\tag{1}$$
seems to approximate
$$\left|\frac …
- 1,001
5
votes
If $\zeta(s)=0$ with $\Re(s)=\frac{1}{2}$, is then $|\hat{\zeta}(s,3)|^2=\frac{1}{2}$?
The intent of this post is to share some results as requested in the comments above and assumes the following definitions from the original question above.
(1) $\quad\zeta(s)=\underset{N\to\infty}{ …
- 1,001
3
votes
The graph and sign of $p_n-\operatorname{ali}(n)$, where $p_n$ is the $n$-th prime and $\ope...
Here's a discrete plot of $p_n-ali(n)$ for $0<n\le 10^9$ in steps of $\Delta n=10^6$ where $ali(n)$ is computed to the nearest $\frac{1}{10}$ .
I used the following Mathematica code to compute $a …
- 1,001
0
votes
Solving a recurrence relation for the prime counting function?
Consider the following fairly simple Mathematica code
Block[{a={a1},n=3},
While[n<=32,
a=AppendTo[a,(PrimePi[n]-Sum[a[[k]] PrimePi[k],{k,1,n-2}])/PrimePi[n-1]];
n++];
a]
which generates the fir …
- 1,001
6
votes
Accepted
Exact formula for partial sums of Liouville function $L(n)$ (OEIS sequence A002819)
$$\lambda(n)=(-1)^{\Omega(n)}=\sum\limits_{d^2|n}\mu\left(\frac{n}{d^2}\right)\tag{1}$$
$$L(x)=\sum\limits_{n\le x}\lambda(n)=\sum\limits_{n^2\le x} M\left(\frac{x}{n^2}\right)\tag{2}$$
where
$$M(x)=\ …
- 1,001
4
votes
"Squeezing" the primes?
This answer describes an alternate approach to "squeezing the primes" (more accurately prime-powers) based on my related Math Overflow question on the inverse Mellin transform
$$p(x)=\mathcal{M}_s^{-1 …
- 1,001