On the remainder term of the prime number formula and the zeros of Riemann’s zeta-function

1984 ◽  
pp. 186-197
Author(s):  
J. Pintz
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Remainder Term ◽  
Number Formula ◽  
Riemann’S Zeta Function

scholarly journals ON LITTLEWOOD’S PROOF OF THE PRIME NUMBER THEOREM

2019 ◽  
Vol 101 (2) ◽  
pp. 226-232
Author(s):  
ALEKSANDER SIMONIČ
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Prime Number Theorem ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
The One ◽  
Riemann’S Zeta Function

In this note we examine Littlewood’s proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the nonvanishing of Riemann’s zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.

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