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Definition df-cht 23568
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 23562 . 2  class  theta
2 vx . . 3  setvar  x
3 cr 9480 . . 3  class  RR
4 cc0 9481 . . . . . 6  class  0
52cv 1397 . . . . . 6  class  x
6 cicc 11535 . . . . . 6  class  [,]
74, 5, 6co 6270 . . . . 5  class  ( 0 [,] x )
8 cprime 14301 . . . . 5  class  Prime
97, 8cin 3460 . . . 4  class  ( ( 0 [,] x )  i^i  Prime )
10 vp . . . . . 6  setvar  p
1110cv 1397 . . . . 5  class  p
12 clog 23108 . . . . 5  class  log
1311, 12cfv 5570 . . . 4  class  ( log `  p )
149, 13, 10csu 13590 . . 3  class  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
)
152, 3, 14cmpt 4497 . 2  class  ( x  e.  RR  |->  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
) )
161, 15wceq 1398 1  wff  theta  =  ( x  e.  RR  |->  sum_
p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p ) )
Colors of variables: wff setvar class
This definition is referenced by:  chtf  23580  chtval  23582
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