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

Definition df-cht 20296
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 20290 . 2  class  theta
2 vx . . 3  set  x
3 cr 8704 . . 3  class  RR
4 cc0 8705 . . . . . 6  class  0
52cv 1618 . . . . . 6  class  x
6 cicc 10625 . . . . . 6  class  [,]
74, 5, 6co 5792 . . . . 5  class  ( 0 [,] x )
8 cprime 12720 . . . . 5  class  Prime
97, 8cin 3126 . . . 4  class  ( ( 0 [,] x )  i^i  Prime )
10 vp . . . . . 6  set  p
1110cv 1618 . . . . 5  class  p
12 clog 19874 . . . . 5  class  log
1311, 12cfv 4673 . . . 4  class  ( log `  p )
149, 13, 10csu 12123 . . 3  class  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
)
152, 3, 14cmpt 4051 . 2  class  ( x  e.  RR  |->  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
) )
161, 15wceq 1619 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  20308  chtval  20310
  Copyright terms: Public domain W3C validator