scholarly journals Explicit bound for the number of primes in arithmetic progressions assuming the generalized Riemann hypothesis

2021 ◽  
Author(s):  
Anne-Maria Ernvall-Hytönen ◽  
Neea Palojärvi
Keyword(s):  
Riemann Hypothesis ◽  
Arithmetic Progressions ◽  
Generalized Riemann Hypothesis ◽  
Explicit Bound ◽  
Primes In Arithmetic Progressions

scholarly journals Explicit short intervals for primes in arithmetic progressions on GRH

2019 ◽  
Vol 15 (04) ◽  
pp. 825-862
Author(s):  
Adrian W. Dudek ◽  
Loïc Grenié ◽  
Giuseppe Molteni
Keyword(s):  
Riemann Hypothesis ◽  
Arithmetic Progressions ◽  
Short Intervals ◽  
Generalized Riemann Hypothesis ◽  
Primes In Arithmetic Progressions

We prove explicit versions of Cramér’s theorem for primes in arithmetic progressions, on the assumption of the generalized Riemann hypothesis.

scholarly journals A mean value of the representation function for the sum of two primes in arithmetic progressions

2017 ◽  
Vol 13 (04) ◽  
pp. 977-990 ◽  
Author(s):  
Yuta Suzuki
Keyword(s):  
Asymptotic Formula ◽  
Riemann Hypothesis ◽  
Mean Value ◽  
Arithmetic Progressions ◽  
Representation Function ◽  
Real Zeros ◽  
Generalized Riemann Hypothesis ◽  
Goldbach Problem ◽  
Primes In Arithmetic Progressions ◽  
The Mean

In this paper, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in arithmetic progressions. This is an improvement of the result of F. Rüppel in 2009, and a generalization of the result of A. Languasco and A. Zaccagnini concerning the ordinary Goldbach problem in 2012.

scholarly journals On the sum of the square of a prime and a square-free number

2016 ◽  
Vol 19 (1) ◽  
pp. 16-24
Author(s):  
Adrian W. Dudek ◽  
David J. Platt
Keyword(s):  
Numerical Computation ◽  
Riemann Hypothesis ◽  
Arithmetic Progressions ◽  
Large Integer ◽  
Primes In Arithmetic Progressions ◽  
Explicit Bounds

We prove that every integer$n\geqslant 10$such that$n\not \equiv 1\text{ mod }4$can be written as the sum of the square of a prime and a square-free number. This makes explicit a theorem of Erdős that every sufficiently large integer of this type may be written in such a way. Our proof requires us to construct new explicit results for primes in arithmetic progressions. As such, we use the second author’s numerical computation regarding the generalised Riemann hypothesis to extend the explicit bounds of Ramaré–Rumely.

Sign in / Sign up

Export Citation Format

Share Document