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Definition df-cht 20746
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 20740 . 2  class  theta
2 vx . . 3  set  x
3 cr 8922 . . 3  class  RR
4 cc0 8923 . . . . . 6  class  0
52cv 1648 . . . . . 6  class  x
6 cicc 10851 . . . . . 6  class  [,]
74, 5, 6co 6020 . . . . 5  class  ( 0 [,] x )
8 cprime 13006 . . . . 5  class  Prime
97, 8cin 3262 . . . 4  class  ( ( 0 [,] x )  i^i  Prime )
10 vp . . . . . 6  set  p
1110cv 1648 . . . . 5  class  p
12 clog 20319 . . . . 5  class  log
1311, 12cfv 5394 . . . 4  class  ( log `  p )
149, 13, 10csu 12406 . . 3  class  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
)
152, 3, 14cmpt 4207 . 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  20758  chtval  20760
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