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Results tagged with prime-numbers
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user 145223
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
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Generalization of Wilson's theorem for prime tuples
We know that Wilson's theorem states the following :
$x$ is a prime if $(\frac {\Gamma(x)+1}{x})$ is an integer .
We can extend this to Twin primes as :
$x$ and $x+2$ is prime if $(\frac {4(\Gamma( …
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0
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0
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List of properties of Twin primes Dirichlet series
In a paper R. Arenstorf - There are infinitely many prime twins
he stated the following Dirichlet series :
$$
T(s) = \sum_{n=1}^\infty \frac{\Lambda(n)\Lambda(n+2)}{n^s}
$$
Question : What are k …
- 774
1
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0
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Validity of analysis of summation of function of primes using Abel–Plana summation:
Consider the analytic function $g(x)$
Define
$$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$$
Note that:
$$f(p)=g(p) \text{ for prime } p$$
And $f(n)=0$ fo …
- 774
1
vote
0
answers
145
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On lacunary series connected with prime number theory
Consider the following lacunary sum with parameter $x$:
$$S(x)=\sum_{n=5}^{\infty}\sin^2\left(\frac{x\Gamma(n)}{n}\right).$$
As we can see for $x=\frac{\pi}{2}$
the sum becomes$$\sum_p\cos^2\left(\fra …
- 774
4
votes
0
answers
921
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Guessing of $n$th prime from "super- regularized" product of primes
( I've been thinking about asking this for a long time . Though this is not rigorous; It can be thought of as heuristic or extraction of information from different viewpoint.)
We know "super-regulariz …
- 774
1
vote
0
answers
473
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Explicit formula for zeta function with special type of weight
Consider the following line of thinking:
$$\pi(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} $$
Here,
$\operatorname{R}(x) = \sum …
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324
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On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometri...
Consider the analytic function $g(x)$
Now define
$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$
Such that
$|f(x+it)|=o(e^{2πt})$
uniformly for every $x …
- 774