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Definition df-cht 20879
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 20873 . 2  class  theta
2 vx . . 3  set  x
3 cr 8989 . . 3  class  RR
4 cc0 8990 . . . . . 6  class  0
52cv 1651 . . . . . 6  class  x
6 cicc 10919 . . . . . 6  class  [,]
74, 5, 6co 6081 . . . . 5  class  ( 0 [,] x )
8 cprime 13079 . . . . 5  class  Prime
97, 8cin 3319 . . . 4  class  ( ( 0 [,] x )  i^i  Prime )
10 vp . . . . . 6  set  p
1110cv 1651 . . . . 5  class  p
12 clog 20452 . . . . 5  class  log
1311, 12cfv 5454 . . . 4  class  ( log `  p )
149, 13, 10csu 12479 . . 3  class  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
)
152, 3, 14cmpt 4266 . 2  class  ( x  e.  RR  |->  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
) )
161, 15wceq 1652 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  20891  chtval  20893
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