MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-cht Unicode version

Definition df-cht 20334
Description: Define the first Chebyshev function, which adds up the logarithms of all primes less than  x. The symbol used to represent this function is sometimes the variant greek letter theta shown here and sometimes the greek letter psi, ψ; however, this notation can also refer to the second Chebyshev function, which adds up the logarithms of prime powers instead. See https://en.wikipedia.org/wiki/Chebyshev_function for a discussion of the two functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
df-cht  |-  theta  =  ( x  e.  RR  |->  sum_
p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p ) )
Distinct variable group:    x, p

Detailed syntax breakdown of Definition df-cht
StepHypRef Expression
1 ccht 20328 . 2  class  theta
2 vx . . 3  set  x
3 cr 8736 . . 3  class  RR
4 cc0 8737 . . . . . 6  class  0
52cv 1622 . . . . . 6  class  x
6 cicc 10659 . . . . . 6  class  [,]
74, 5, 6co 5858 . . . . 5  class  ( 0 [,] x )
8 cprime 12758 . . . . 5  class  Prime
97, 8cin 3151 . . . 4  class  ( ( 0 [,] x )  i^i  Prime )
10 vp . . . . . 6  set  p
1110cv 1622 . . . . 5  class  p
12 clog 19912 . . . . 5  class  log
1311, 12cfv 5255 . . . 4  class  ( log `  p )
149, 13, 10csu 12158 . . 3  class  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
)
152, 3, 14cmpt 4077 . 2  class  ( x  e.  RR  |->  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
) )
161, 15wceq 1623 1  wff  theta  =  ( x  e.  RR  |->  sum_
p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p ) )
Colors of variables: wff set class
This definition is referenced by:  chtf  20346  chtval  20348
  Copyright terms: Public domain W3C validator