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1
Content available Deformation of semicircular and circular laws via p-adic number fields and sampling of primes
Cho Ilwoo, Jorgensen Palie E. T.
Opuscula Mathematica
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2019
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Vol. 39, no. 6
773--813
EN
In this paper, we study semicircular elements and circular elements in a certain Banach *-probability space [formula] induced by analysis on the p-adic number fields Qp over primes p. In particular, by truncating the set P of all primes for given suitable real numbers t < s in R, two different types of truncated linear functionals [formula], and [formula] re constructed on the Banach *-algebra [formula]. We show how original free distributional data (with respect to r°) are distorted by the truncations on P (with respect to [formula], and [formula]). As application, distorted free distributions of the semicircular law, and those of the circular law are characterized up to truncation.
2
Content available Semicircular elements induced by p-adic number fields
Cho I., Jorgensen P. E. T.
Opuscula Mathematica
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2017
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Vol. 37, no. 5
665--703
EN
In this paper, we study semicircular-like elements, and semicircular elements induced by p-adic analysis, for each prime p. Starting from a p-adic number field Qp, we construct a Banach *-algebra [formula], for a fixed prime p, and show the generating elements Qpj of [formula] form weighted-semicircular elements, and the corresponding scalar-multiples Θpj of Qpj become semicircular elements, for all j ∈ Z. The main result of this paper is the very construction of suitable linear functionals [formula] on [formula], making Qpj be weighted-semicircular, for all j ∈ Z.
3
Content available Weryfikacja „słabej” hipotezy Goldbacha do 1031
Świerczewski Ł.
Biuletyn Naukowy Wrocławskiej Wyższej Szkoły Informatyki Stosowanej. Informatyka
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2013
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vol. 3
28--31
PL
Praca prezentuje aspekt numerycznej weryfikacji „słabej” hipotezy Goldbacha dla wartości mniejszych niż 1031. Do obliczeń, które zajęły w sumie ok. 50 000 godzin czasu pojedynczego CPU wykorzystano klaster wydajnościowy złożony z procesorów AMD Opteron 4284. Podczas sprawdzania pierwszości zastosowano test Millera-Rabina. Przetestowano także możliwe zastosowanie testu ECPP. Jak się okazało przy założeniu dodatkowych warunków poprawności testu Millera-Rabina „słaba” hipoteza Goldbacha w badanym zakresie jest prawidłowa.
EN
This paper presents aspect of the numerical verification a „weak” Goldbach’s conjecture for values less than 1031. For calculations, that took about 50 000 hours of a single CPU performance, there was used an performance cluster consisting of the AMD Opteron 4284 processors. During the primality check, there was used Miller-Rabin test. There was also tested the possiblity of ECPP test usage. As it turned out, when there were added some additional conditions of correctness of Miller-Rabin test, the „weak” Goldbach’s conjecture occurs correct in researched range.
4
Content available remote The Skewes Number for Twin Primes: Counting Sign Changes of pi 2(x) - C2Li2(x)
Wolf M.
Computational Methods in Science and Technology
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2011
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Vol. 17, No. 1-2
87-92
EN
The results of computer investigation of the sign changes of the difference between the number of twin primes pi2 (x) and the Hardy-Littlewood conjecture C2Li2 (x) are reported. It turns out that d2 (x) = pi2 (x) - C2Li2 (x) changes the sign at unexpectedly low values of x and for x less than 248 = 2.81... x 1014 there are 477118 sign changes of this difference. It is conjectured that the number of sign changes of d2 (x) for x element of (1, T ) is given by T log(T). The running logarithmic densities of the sets for which d2 (x) greather than 0 and d2 (x) less than 0 are plotted for x up to 2 48.
5
Content available remote On least prime primitive roots mod 2p for odd primes p
Paszkiewicz A.
Electronics and Telecommunications Quarterly
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2009
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Vol. 55, No 1
57-69
EN
In this paper we derive a conditional formula which allows to compute the natural density of prime numbers with a given least prime primitive root modulo 1p and compare theoretical results with the numerical evidence. We also illustrate graphically these densities as functions of the upper limit x for primes below x.
6
Content available remote Numerical investigation of least prime primitive roots modulo a prime number
Paszkiewicz A.
Biuletyn Wojskowej Akademii Technicznej
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1999
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Vol. 48, nr 10
41-52
EN
For each of the 105,097,564 primes p from the interval [3,2 31] there exists a primitive root h(p)
PL
Dla każdej spośród 105,007,564 liczb pierwszych p z przedziału [3,2 31] istnieje pierwiastek pierwotny h(p)
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