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Definition df-cht 22434
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 22428 . 2  class  theta
2 vx . . 3  setvar  x
3 cr 9281 . . 3  class  RR
4 cc0 9282 . . . . . 6  class  0
52cv 1368 . . . . . 6  class  x
6 cicc 11303 . . . . . 6  class  [,]
74, 5, 6co 6091 . . . . 5  class  ( 0 [,] x )
8 cprime 13763 . . . . 5  class  Prime
97, 8cin 3327 . . . 4  class  ( ( 0 [,] x )  i^i  Prime )
10 vp . . . . . 6  setvar  p
1110cv 1368 . . . . 5  class  p
12 clog 22006 . . . . 5  class  log
1311, 12cfv 5418 . . . 4  class  ( log `  p )
149, 13, 10csu 13163 . . 3  class  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
)
152, 3, 14cmpt 4350 . 2  class  ( x  e.  RR  |->  sum_ p  e.  ( ( 0 [,] x )  i^i  Prime ) ( log `  p
) )
161, 15wceq 1369 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  22446  chtval  22448
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