+Σ r=1. r p. ( p,xr) - Σ. ( p ) r p

Transcription

1 5 Formulas for Rimann Zta at coml numbr W can tnd " Formulas for Rimann Zta at natural numbr " to th coml numbr asily. 5.1 Formulas of coth family Formula such that 1,0,,-, and (, ) = t -t dt is incomlt gamma function,, u0. 1 (,r) - B r ( ) = () - ( ) r +r r( +r) +1 ( ) = ( ) ( +1) (,r) ( ) - B r r r( +r) W obtaind th following Rimann Zta in Formula.1.1 in " Formulas for Rimann Zta at natural numbr ". ( n ) = ( )! 1 - n n ( r) s s! 1 - r n r -n B r n+r r r( +r)! On th othr hand, for th rlation btwn th sum of onntial and th incomlt gamma function, th following formula is known. s = ( n,) ( n) Substituting this for th abov, ( n ) = ( )! 1 - n n ( n,r) ( n) r n -n B r n+r r r( +r)! Etnding natural numbr n to coml numb and rlacing th factorial with th gamma function, w obtain B r +r ( ) = () 1 - (,r) ( ) Eaml ( 1/ i ) - r r( +r) Whn =1+i, this is calculatd according to th formula. As th rsult of calculating th sris to th 55 th trm, th significant 3 digits wr obtaind

2 In a similar way, w obtain th following formula from Formula.1.1' (.1 ). Formula 5.1.1' such that 1,0,,-, and (, ) = t -t dt is incomlt gamma function,, u0. 1 ( ) ( ) = () + -log + +1 ( ) = ( ) ( +1) + (,r) ( ) (,r) ( ) Eaml ( 1/ i ) 1- B r r +r r( +r) 1- B r r r( +r) Whn =1-i, this is calculatd according to th formula. As th rsult of calculatingth sris to th 51 th trm, th significant digits wr obtaind. - -

3 5. Formula of tanh family Formula 5..1 such that 1,0,,-, and (, ) = t -t dt is incomlt gamma function,, u0. ( ) = (,r) () ( +1) r - ( ) r r r B r +r r( +r) ( ) = 1 ( +1) (,r) () r - ( ) r r r B r r( +r) W obtaind th following Rimann Zta in Formula..1 in.. ( n ) = n Substituting for this, ( r) s = ( n ) = n n n n! ( r) s () r -n r n r r r B r n+r r( +r)! ( n,r) ( n) r n n n! ( n,r) () r -n ( n) r n r r B r n+r r( +r)! Etnding natural numbr n to coml numb and rlacing th factorial with th gamma function, w obtain th dsird rssion. Eaml ( 0.7) Whn =1/+i, this is calculatd according to th formula. As th rsult of calculating th sris to th 37 th trm, th significant 10 digits wr obtaind

4 5.3 Formulas of csch family Formula such that 1,0,,-, and (, ) = t -t dt is incomlt gamma function,, u0. ( ) = ( ) = ( ) ( ) 1 ( ) ( ) W obtaind th following Rimann Zta in Formula.3.1 in.3. ( n ) = On th othr hand, Thn, s = n ( )!( ) ( n,) ( n) ( r) s = Substituting this for th abov, ( n ) = n,( r) ( )( r) - r -B r +r r r( +r) (,r) ( )( r) r -B r r r( +r) n,( r) ( r) ( n) n ( )!( ) n 1 + -( r) ( r) s ( r) n -n r -B r n+r r r( +r)! n,( r) ( n )( r) n -n r -B r n+r r r( +r)! Etnding natural numbr n to coml numb and rlacing th factorial with th gamma function, w obtain th dsird rssion. In a similar way, w obtain th following formula from Formula.3.1' (.3 ). Formula 5.3.1' - 4 -

5 such that 1,0,,-, and (, ) = t -t dt is incomlt gamma function,, u0. ( ) = ( ) Escially whn =1,, 1 ( ) = ( ) 1 -log ( ) = ( ) ( ) Eaml (i ) log - 1- r 1 +,( r) ( )( r) r -B r +r r( ) +r (,r) ( )( r) 1- r (,4r-) ( )( r) r r -B r r( +r) r( ) +r r -B r +r According to th formula at = this is calculatd. As th rsult of calculating th sris to th 18 th trm, th significant 10 digits wr obtaind Alin's Mathmatics K. Kono - 5 -