Exploring Prime Numbers and Modular Functions III

1 downloads 0 Views 486KB Size Report
Exploring Prime Numbers and Modular Functions III: On the Exponential of Prime Number via ... Dr. K. Raja Rama Gandhi2. Number Theorist, Brazil1. Resource ...
Bulletin of Mathematical Sciences and Applications ISSN: 2278-9634, Vol. 7, pp 34-37 doi:10.18052/www.scipress.com/BMSA.7.34 © 2014 SciPress Ltd., Switzerland

Online: 2014-02-03

Exploring Prime Numbers and Modular Functions III: On the Exponential of Prime Number via Dedekind Eta Function Edigles Guedes1 and Prof. Dr. K. Raja Rama Gandhi2 Number Theorist, Brazil1 Resource perosn in Mathematics for Oxford University Press and Professor at BITS-Vizag2

ABSTRACT. The main objective this paper is to develop asymptotic formulas for the exponential of prime number, using Dedekind eta function. 1. INTRODUCTION As consequence of the prime number theorem, I get the asymptotic formula for the nth prime number, denoted by (1) M. Pervouchine, in Mémories de la Société physic-mathématique de Kasan, [1, page 848] deduced that (2) On the other hand, Ernest Cesáro, in Sur une formule empirique de M. Pervouchine [1, page 849], I encounter the formula (

) (

(3)

)

in modern notation, (

) (

((

)

)

)

(4)

On the other hand, the Dedekind eta function was introduced by Richard Dedekind, in 1877, { ( ) } by the equation [2, page 47] and is defined in the half-plane ⁄

( )



The infinite product has the form ∏( the product converges absolutely and is nonzero.

(

)

) where

(5) then | |

If

so

In [2, page 48], I have







(

)

(6)

(

)

(7)

Hence,







and







SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/

(8)

Bulletin of Mathematical Sciences and Applications Vol. 7

35

for In this paper, I prove, among other things, that (

(

(

(

)

)(

)

(

)



)[ (

)

)





]



(

)

2. THEOREM LEMMA 1. I have ( where

)

(

)

denotes the nth prime number defined by

Proof. Multiplying (3) by ( (

)

(

)

(

) I obtain )

thereupon, letting

(

)

(

)

(

)

I find ( (

)

)

(

(

)

)

COROLLARY 1. I have ⌊

⌊ where



(



)



THEOREM 1. For

sufficiently large, then

and ⌊ ⌋ denotes the floor function.

arising from the definition of the floor function.

[



)









( where

)

denotes the nth prime number defined by

Proof. I note that ⌊

[ (

(

denotes the nth prime number.

)



]{ [

][ (

(

(

)

)]

)

(

)

]}

36

Volume 7

Proof. I put (7) and (8) into Lemma 1, as following [



(

](





)[ ∑

)





(

)]

Dividing both members of above equation by (

)

[







[ ∑ I return to replace





by

]( (

[ (



[ ∑

]



][



(

Multiplying both members of (9) by [



[ ∑

)

)





[ (

)]

)

( ]

(

)









)]

in the previous equation (

[

)

(

)

]

)

(9)

)

I encounter ][ (

(

(

(

)] { [

(

(

)

)]

)

(

)

]}

)

consequently, [

[ (



)









( This completes the Theorem.

)



]{ [

][ (

(

(

)

)]

)

(

)

]}

Bulletin of Mathematical Sciences and Applications Vol. 7

COROLLARY 2. For

sufficiently large, then

(

( where

(

37

(

)

(

)(

)

)[ (

)



)



)



]



(

)

denotes the nth prime number.

Proof. I note that, for



(11)

From Theorem 1 and (11), I obtain (

(

(

(

)

(

)

)(

)[ (

)



)



)



]



(

)

This completes the proof. CONJECTURE: I observe that, numerically,

is better approximate if I add the term

so (

(

(

(

)

(

)

)

)(

)[ (



)



)





]

(

)

REFERENCES [1] Cesàro, Ernest, Sur une formule empirique de M. Pervouchine, Comptes rendus hebdomadaires des séances de l’Academia des sciences 119: 848-849, (1894). (French) [2] Apostol, Tom M., Modular functions and Dirichlet series in number theory, Springer Verlag, 2000. καιδιζlκαιs