scholarly journals ON MIXED JOINT DISCRETE UNIVERSALITY FOR A CLASS OF ZETA-FUNCTIONS: A FURTHER GENERALIZATION

2020 ◽  
Vol 25 (4) ◽  
pp. 569-583
Author(s):  
Roma Kačinskaitė ◽  
Kohji Matsumoto
Keyword(s):  
Linear Independence ◽  
Zeta Functions ◽  
Prime Numbers ◽  
Value Distribution ◽  
Arithmetic Progressions ◽  
Independence Condition ◽  
Discrete Universality

We present the most general at this moment results on the discrete mixed joint value-distribution (Theorems 5 and 6) and the universality property (Theorems 3 and 4) for the class of Matsumoto zeta-functions and periodic Hurwitz zeta-functions under certain linear independence condition on the relevant parameters, such as common differences of arithmetic progressions, prime numbers etc.

scholarly journals On mixed joint discrete universality for a class of zeta-functions. II

2019 ◽  
Vol 59 (1) ◽  
pp. 54-66
Author(s):  
Roma Kačinskaitė ◽  
Kohji Matsumoto
Keyword(s):  
Zeta Functions ◽  
Discrete Universality

About finding of prime numbers that follows after given prime number without using computer

2020 ◽  
pp. 81-85
Author(s):  
V. S. Malakhovsky
Keyword(s):  
Prime Number ◽  
Prime Numbers ◽  
Arithmetic Progressions

It is shown how to define one or several prime numbers following af­ter given prime number without using computer only by calculating sev­eral arithmetic progressions. Five examples of finding such prime num­bers are given.

Almost-primes in arithmetic progressions and short intervals

1978 ◽  
Vol 83 (3) ◽  
pp. 357-375 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Short Intervals ◽  
Almost Primes ◽  
Primes In Arithmetic Progressions ◽  
Image Position

In this paper we shall investigate the occurrence of almost-primes in arithmetic progressions and in short intervals. These problems correspond to two well-known conjectures concerning prime numbers. The first conjecture is that, if (l, k) = 1, there exists a prime p satisfying

Hua’s theorem on sums of five prime squares in arithmetic progressions

2008 ◽  
Vol 45 (1) ◽  
pp. 29-66
Author(s):  
Claus Bauer
Keyword(s):  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Prime Squares ◽  
Positive Integers

It is proved that for a given integer N and for all but ⪡ (log N ) B prime numbers k ≦ N5/96 − ε the following is true: For any positive integers bi , i ∈ {1, 2, 3, 4, 5}, ( bi , k ) = 1 that satisfy N ≡ b12 + b22 + b32 + b42 + b52 (mod k ), N can be written as N = p12 + p22 + p32 + p42 + p52 , where the pi , i ∈ {1, 2, 3, 4, 5} are prime numbers that satisfy pi ≡ bi (mod k ).

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