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Definition df-cht 20330
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 20324 . 2  class  theta
2 vx . . 3  set  x
3 cr 8733 . . 3  class  RR
4 cc0 8734 . . . . . 6  class  0
52cv 1624 . . . . . 6  class  x
6 cicc 10655 . . . . . 6  class  [,]
74, 5, 6co 5821 . . . . 5  class  ( 0 [,] x )
8 cprime 12754 . . . . 5  class  Prime
97, 8cin 3154 . . . 4  class  ( ( 0 [,] x )  i^i  Prime )
10 vp . . . . . 6  set  p
1110cv 1624 . . . . 5  class  p
12 clog 19908 . . . . 5  class  log
1311, 12cfv 5223 . . . 4  class  ( log `  p )
149, 13, 10csu 12154 . . 3  class  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
)
152, 3, 14cmpt 4080 . 2  class  ( x  e.  RR  |->  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
) )
161, 15wceq 1625 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  20342  chtval  20344
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