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Definition df-cht 20832
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 20826 . 2  class  theta
2 vx . . 3  set  x
3 cr 8945 . . 3  class  RR
4 cc0 8946 . . . . . 6  class  0
52cv 1648 . . . . . 6  class  x
6 cicc 10875 . . . . . 6  class  [,]
74, 5, 6co 6040 . . . . 5  class  ( 0 [,] x )
8 cprime 13034 . . . . 5  class  Prime
97, 8cin 3279 . . . 4  class  ( ( 0 [,] x )  i^i  Prime )
10 vp . . . . . 6  set  p
1110cv 1648 . . . . 5  class  p
12 clog 20405 . . . . 5  class  log
1311, 12cfv 5413 . . . 4  class  ( log `  p )
149, 13, 10csu 12434 . . 3  class  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
)
152, 3, 14cmpt 4226 . 2  class  ( x  e.  RR  |->  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
) )
161, 15wceq 1649 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  20844  chtval  20846
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