^^Zeta function. The Riemann hypothesis. Euler Prime Product Formula.

1  +  1

2

  +  1
3		  +  1
4		  +  1
5		  + ...    ≡     ∑
n≥1		   1
n		      

serie armonica
                               
1  +  1

22

  +  1
32		  +  1
42		  +  1
52		  + ... ≡  ∑
n≥1		   1
n2		  

∑ 1/n²  Basel problem
                               
1  +  1

2s

  +  1
3s		  +  1
4s		  +  1
5s		  + ... ≡  ∑
n≥1		   1
ns		  

ζ(s) zeta function

Euler product formula. wp

zeta
function		  ζ(s) ≡  ∑
n≥1		   1
ns		    =   ∏
p prime		   1
1-p-s

 

1  +  1

2s

  +  1
3s		  +  1
4s		  +  1
5s		  + ...   =   1

1-2-s

  ·  1
1-3-s		  ·  1
1-5-s		  ·  1
1-7-s		  + ...

 

1

1-2-s

  ·  1
1-3-s		  ·  1
1-5-s		  ·  1
1-7-s		  ·  1
1-11-s		  + ...  ≡   ∏
p prime		   1
1-p-s
                           

Bernhard Riemann defined zeta function and proved its basic properties

in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude"

"Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse"

proofwiki/Sum_of_Sequence_of_Squares

Dirlo

  1. Prime Product Formula
  2. Euler Prime Product Formula
  3. Euler product (of the primes)

is the main bridge that connects prime numbers and Zeta function

Links

  1. yt/Riemann Hypothesis - Numberphile
  2. yt/The Key to the Riemann Hypothesis - Numberphile
  3. yt/Visualizing the Riemann hypothesis and analytic continuation 3Blue1Brown
  4. yt/gen2021 Alex Kontorovich, professor of mathematics at Rutgers University
  5. yt/What is the Riemann Hypothesis? Physics Explained
  6. yt/700 years of secrets of the Sum of Sums (paradoxical harmonic series) Mathologer
  8. yt/Euler Finds The Prime Product Formula
  9. yt/Euler’s Pi Prime Product and Riemann’s Zeta Function Mathologer
  10. yt/Factorials, prime numbers, and the Riemann Hypothesis zetamath
  12. wp/Euler_product_formula
  13. wp/Prime-counting_function
  14. wp/Prime_number_theorem
  16. claymath/riemanns-1859-manuscript
  17. terrytao/2021/the-riemann-zeta-function-and-the-prime-number-theorem

 

Talk

Studio espo simbolo sommatoria con indici

1  +  1

2

  +  1
3		  +  1
4		  +  1
5		  +  1
6		  + ... =  ?        ∑   1
n		 indice  
cella sotto   
                          n≥1      
1  +  1

2

  +  1
3		  +  1
4		  +  1
5		  +  1
6		  + ... =  ?        ∑
n≥1		   1
n		 riga sotto
normal
                                 
1  +  1

2

  +  1
3		  +  1
4		  +  1
5		  +  1
6		  + ... =  ?        ∑
n≥1		   1
n		 sub
                                 
1  +  1

2

  +  1
3		  +  1
4		  +  1
5		  +  1
6		  + ... =  ?        ∑
n≥1		   1
n		 sup
                                 
1  +  1

2

  +  1
3		  +  1
4		  +  1
5		  +  1
6		  + ... =  ?        ∑

n≥1

   1
n		 50%
                                 

 

 

1  +  1

2

  +  1
3		  +  1
4		  +  1
5		  +  1
6		  + ... =  ?        ∑
n≥1		   1
n		   n>1 
                                 
1  +  1

22

  +  1
32		  +  1
42		  +  1
52		  +  1
62		  + ... =  ?    ∑   1
n2		   n>1 
                                 
1  +  1

23

  +  1
33		  +  1
43		  +  1
53		  +  1
63		  + ... =  ?    ∑   1
n3		   n>1 
                                 
1

1s

  +  1

2s

  +  1
3s		  +  1
4s		  +  1
5s		  +  1
6s		  + ... =  ?    ∑   1
ns		   n>1 
                                 
                                 
1s  +  2s  +  3s  +  4s  +  5s  +  6s  + ... =  ?    ∑   ns   n>1 

 

                                             
1
  +  1

2s

  +  1
3s		  +  1
4s		  +  1
5s		  +  1
6s		  + ...   =   1

1-2-s

  ·  1
1-3-s		  ·  1
1-5-s		  ·  1
1-7-s		  ·  1
1-11-s		  + ...
                                             

 

1  +  1

2

  +  1
3		  +  1
4		  +  1
5		  +  1
6		  + ...    ≡     ∑
n≥1		   1
n		      

serie armonica

     
                                   
1  +  1

2s

  +  1
3s		  +  1
4s		  +  1
5s		  +  1
6s		  + ... ≡  ∑
n≥1		   1
ns		  

ζ(s) zeta function
                                  ∑ 1/n²  Basel problem