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Definition df-cht 20350
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 20344 . 2  class  theta
2 vx . . 3  set  x
3 cr 8752 . . 3  class  RR
4 cc0 8753 . . . . . 6  class  0
52cv 1631 . . . . . 6  class  x
6 cicc 10675 . . . . . 6  class  [,]
74, 5, 6co 5874 . . . . 5  class  ( 0 [,] x )
8 cprime 12774 . . . . 5  class  Prime
97, 8cin 3164 . . . 4  class  ( ( 0 [,] x )  i^i  Prime )
10 vp . . . . . 6  set  p
1110cv 1631 . . . . 5  class  p
12 clog 19928 . . . . 5  class  log
1311, 12cfv 5271 . . . 4  class  ( log `  p )
149, 13, 10csu 12174 . . 3  class  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
)
152, 3, 14cmpt 4093 . 2  class  ( x  e.  RR  |->  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
) )
161, 15wceq 1632 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  20362  chtval  20364
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