On Relation Of The Riemann Zeta Function To Its Partial Product De…nitions

On Relation Of The Riemann Zeta Function To Its Partial Product De…nitions

Vahid Rahmati

Received 16 August 2019

Abstract

New relations for the Riemann zeta function (RZF) by de…ning supplementary partial product func- tions are developed in this paper. Relations are based on partial products of prime numbers with recourse to product form of the RZF found by unique factorization in Z. This paper is in pursuit of generat- ing new identities— involving RZF including summations, products, and limits— mostly in the matter of multiplicative property of Euler products and by applying Taylor series. This is done with the intention of relating some classical and newly de…ned functions (using multiplicative Jordan’s totient function and primorial sequence) to RZF in the form of theorems and proofs.

1 Introduction

The famous Riemann zeta function (s)(RZF) has a variety of applications in mathematics, in particular, in

…eld of number theory which makes it an important special function. In addition to its direct applications, there exist several open problems in the …eld solvable in the event of a correct proof of the well-known Riemann hypothesis concerning real part of non-trivial zeros located in vertical lineR(s) =¹₂. The Riemann hypothesis proof as a key towards answers to many problems can be found by equivalences which transform di¢ culty level of the problem into a new level [7].

The paper is organized as 2 main sections: 1- A brief introductory section (2) for RZF 2- Section 3 for developing new formulas, relations, and theorems with 2 subsections concerning RZF as Taylor series coe¢ cients in subsection 3.1, and other special sums, products, and limits for RZF in subsection 3.2. Section 2 provides fundamental, classical de…nitions involving RZF and other related number theory functions for the purpose of an overview.

The development of new relations for RZF is done by de…ning speci…c auxiliary functions, named and introduced as lemmas, in order to acquire new series and limit-based formulas for this function. However, other identities for RZF are developed by consideration of partial product of a special case of Euler products.

The method is also useful when a relation between primorial function and RZF is needed.

2 Riemann zeta function

The RZF de…nition and its relation to L-functions with its product expansion are reviewed brie‡y in this part.

De…nition 1 (Riemann zeta function [1]) The RZF is de…ned by (s) =

X1 n=1

1

n^s (1)

forR(s)>1.

Mathematics Sub ject Classi…cations: 11M06, 11M41, 33E20, 11R42.

yAyandeh Samin Electronics & ICDST, Tehran, Iran

388

Now, with respect toL-functions [1], introduced by Dirichlet (1837), which is de…ned by L(s; ) =

X1 n=1

(n)

n^s ; R(s)>1; (2)

relation (1) is written as

(s) =L(s;1): (3)

Product expansion of (1) [1] is de…ned as (s) =Y

p

1

1 p ^s; R(s)>1; (4)

where product is taken over all primes. Identity (4) is a special case of product of theL-functions where is considered as a multiplicative function [16]. Equation (4) is product over all prime numbers, therefore it gives

0(s)

(s) = X1 n=1

(n)n ^s; R(s)>1 (5)

calculated by taking logarithmic derivative. In (5), (n)is the Mangoldt’s function [1] which equals zero for n6=p^r, and it equals to

(n) = log(p) forn=p^r wherer2N.

3 Riemann Zeta Function in Taylor Series and Some Special Series

With an eye to other formulas involving RZF, and in contrast to previous section showing some classic relations for RZF, in this part, some identities involving RZF including sums and products are presented.

Majority of these formulas are related to RZF as coe¢ cients in power series of di¤erent functions.

3.1 Taylor Series Involving RZF and Other Special Functions

The main intention is to develop power series with function separable in coe¢ cients. Assume that left-hand side of following relation admits Taylor series expansion in neighborhood ofx₀ in the form of

X

n 1

f(x=n^p)

n =X

n 0

f⁽ⁿ⁾(x₀)

n! (x x₀)ⁿ ( +np); (6)

which is found by applying Taylor series to left-side of equation and is valid forf^(m)(x0) =f^(m)(x0=2^p) =:::

where m = 0;1; : : :. In other words, these parts of coe¢ cients should be factored for the zeta function to appear. Variablep shifts argument in right side, and condition for which (1)not appearing in right side is -1 np for andpas integers. This shifting of variable is applicable on argument of zeta function in series presented in this paper, hence it is not re-mentioned in next examples.

Ifx= 1with coe¢ cientsg(n) =f(1=n), left hand side of (6) is aDirichlet-series shown as G(s) =

X1 n=1

g(n)n ^s;

where (s) =G(s)forg(n) = 1. As an example,f(x) = sin(x x₀)is standard form off in (6) to be valid.

Equation (6) is reformulated as X

n 1

f((x x0)=n^p)

n =X

n 0

f⁽ⁿ⁾(0)

n! (x x₀)ⁿ ( +np)

to take functions likef(x) = sin(x)as well provided that zeta function is separable in coe¢ cients as in (6).

However, in this paper form of equation (6) is selected for generated Taylor series centered atx0 = 0. If argument off is inverted and xset to1, (6) forms a generalized Euler sum which is itself a special case of Dirichlet-series [2], [3], represented as

H(s) = X1 n=1

h(n)

n^s ; R(s)>1;

for various functions in place ofh(n)which is de…ned in [15]. However, by inverting argument off, right-side of (6) forms aLaurent series. Ifh(n)is aDirichlet character(mod k),H(s) =L(s; )becomes meromorphic continuation of it [8], [14].

Example 1 Forf(x) = sin(x)near x0= 0,(6) yields X

n 1

sin(x=n^p)

n =X

n 0

( 1)ⁿx²ⁿ⁺¹

(2n+ 1)! ( + (2n+ 1)p):

Next, Lemma1 is introduced:

Lemma 1 The de…ned Upsilon-function

p( ) =X

n 0

n^p

nn!; 6= 0; (7)

where forp= 0it gives

0( ) =e¹; can be calculated for p= 1;2; :::by recursive relation

p( ) = ⁰p 1( ):

Proof. This result is provable by induction.

In view of _p( ), assume following series expansion X

n 0

f(nx)

nn! =X

n 0

f⁽ⁿ⁾(x₀)

n! (x x₀)ⁿ _n( ); 6= 0; (8)

which is valid if f^(m)(x₀) = f^(m)(2x₀) = ::: for m = 0;1; : : :. By applying (6)-(8), following double sum involving function is found as

X

m 1

X

n 0

f(nx=m^p)

m ⁿn! =X

n 0

f⁽ⁿ⁾(x0)

n! (x x0)ⁿ n( ) ( +np); 6= 0: (9) Next, some examples with Lemma2are presented.

Example 2 Sums

X

n 0

sin(nx)

n = sin(x)

2+ 1 2 cos(x); j j>1 (10)

and X

n 0

cos(nx)

n =

2 cos(x)

2+ 1 2 cos(x); j j>1 (11)

have closed form expressions evaluated using Euler identities for sin(x)andcos(x)functions. Integration of (10) is calculable explicitly after removing …rst zero term from left-side and dividing¹ right-side nominator and denominator integrand by ² as

X

n 1

cos(nx)

n ⁿ =

Z ₁ sin(x)

1 + ² 2 ¹cos(x)dx=1

2ln(1 + 1

2

2cos(x)): (12)

Integration of (12)gives X

n 1

sin(nx) n^{2 n} =

Z x 0

1

2ln(1 + 1

2

2cos( ))d

=1= Z x

0

ln(2 sin(x

2))dx (13)

which is a generalized form for the Clausens’s integral [10] originally de…ned for = 1. Another relation is cosine series [10] of the form

X

n 1

cos(nx) n² = x²

4 x 2 +

2

6 ;

which is equal to left-hand side of (11)for = 1after integrating twice [10].

For integration of (11), Lemma2 is introduced:

Lemma 2 The de…ned sigma-function

&p( ) =X

n 0

n^p

n; j j>1; (14)

where forp= 0it equals

&0( ) = 1; is calculated forp= 1;2; :::by recursive relation

&_p( ) = &⁰_{p 1}( ):

Proof. This result is provable by induction.

Sigma-function is related to the Polylogarithm-function by following relation:

&m( ) = ( 1)^m+1Li m( ); m= 0;1;2; :::: (15) A formula similar to (8) using Lemma2is expansion

X

n 0

f(nx)

n =X

n 0

f⁽ⁿ⁾(x0)

n! (x x0)ⁿ&n( ): (16)

The generating function for&n( )using (16) wheref(x) =e^x is found as 1

1 ¹e^x =X

n 0

xⁿ

n!&n( ); x <lnj j:

For = 1, (16) diverges according to (14), therefore, using relationLim( 1) = (m)(2^{1 m} 1)extracted based on Hardy’s series for RZF [11],

(s) = 1 1 2^{1 s}

X1 n=1

( 1)^{n 1}

n^s ; R(s)>0;

1The integrand is reformed according to desired domain in order to result in correct integration as for right-side of (10), j j<1is part of domain where left-side diverges, thus, the integrand was changed to …t left-side domain as well.

wherem6= 1 with (15), an alternative series results in X

n 0

( 1)ⁿf(nx) =X

n 0

f⁽ⁿ⁾(x0)

n! (x x0)ⁿ( 1)ⁿ⁺¹Li n( 1): (17) Moreover, (17) is reducible to

X

n 0

( 1)ⁿf(nx) = 1

2f(x₀) +X

n 0

f⁽²ⁿ⁺¹⁾(x0)

(2n+ 1)! (x x₀)²ⁿ⁺¹ ( 2n 1)(4ⁿ⁺¹ 1); (18) where even powers— in general form of series— are removed by ( 2n) = 0and …rst term equals (0)f(x₀).

To rewrite series in terms of the Bernoulli-numbers [1], ( n)should be replaced by ^B_n+1ⁿ⁺¹. Many diver- gent series are evaluated using various methods including Ramanujan’s summation. In fact, Ramanujan’s summation equals (18) with little variations forx0= 0andx= 1in terms ofBernoulli-numbers [6].

Example 3 By (16)where f(x) = cos(x), series expansion of (real part) integration of right-hand side of (11)is evaluated by separating …rst term as

x+X

n 1

sin(nx)

n ⁿ =

Z 1 ¹cos(x)

1 + ² 2 ¹cos(x)dx=X

n 0

( 1)ⁿ

(2n+ 1)!x²ⁿ⁺¹&_2n( ); (19) forj j 1. The power series for (10)and (11)are equivalent to

sin(x)

2+ 1 2 cos(x) = X1 n=0

( 1)ⁿx²ⁿ⁺¹

(2n+ 1)! &_2n+1( )

and

2 cos(x)

2+ 1 2 cos(x) = X1 n=0

( 1)ⁿx²ⁿ (2n)! &2n( ) forj j>1.

A new double sum formula similar to (9) is found by Lemma2 as X

m 1

X

n 0

f(nx=m^p)

m ⁿ =X

n 0

f⁽ⁿ⁾(x0)

n! (x x0)ⁿ&n( ) ( +np); >1: (20) Next examples illustrate use of formula (20).

Example 4 Letting f(x) = sin(x)and f(x) = cos(x)withp= 1 in (20)using previous formulas gives X

m 1

sin(x=m)

m ( ²+ 1 2 cos(x=m))=X

n 0

( 1)ⁿ

(2n+ 1)!x²ⁿ⁺¹&2n+1( ) (2n+ + 1); >1

and X

m 1

2 cos(x=m)

m ( ²+ 1 2 cos(x=m)) =X

n 0

( 1)ⁿ

(2n)!x²ⁿ&2n( ) (2n+ ); >1:

To develop more formulas in following, Lemma3 is introduced:

Lemma 3 The de…ned tau-function

p( ) =X

n 1

1

n^{p n}n!; 6= 0 (21)

where forp= 0it equals

0( ) =e¹ 1;

is calculated forp= 1;0; :::by recursive relation

p( ) = Z

p 1( ) d :

Proof. This result can be veri…ed by induction.

By using Lemma3, series for left-side of equation (22) is assumed as its right-side:

X

n 1

f(nx) n^{p n}n! =X

n 0

f⁽ⁿ⁾(x0)

n! (x x0)ⁿ p n( ): (22)

For to appear, double summation method applied on (22) as before results in X

m 1

X

n 1

f(nx=m^p1) m n^{p n}n! =X

n 0

f⁽ⁿ⁾(x0)

n! (x x0)ⁿ p n( ) ( +np1); >1: (23) Tau function satis…es n(1) =eB(n)forn >1, whereB(n)is theBell-number [4], [5] appearing in series [9] of the form

X

n 0

f(nx)

n! =eX

n 0

f⁽ⁿ⁾(x₀)

n! (x x₀)ⁿB(n): (24)

Next example illustrates a property of these numbers.

Example 5 Assume that a set ofm+ 1recurrence functions de…ned as

f₀(x) =X

n 0

an

n!(x x₀)ⁿ;

fm(x) =X

n 0

fm 1(nx) n! ;

exists. Then, using Bell-numbers, the Taylor series expansion offm(x)near x0 form2Ris equal to

f_m(x) =e^mX

n 0

an

n!(x x₀)ⁿB^m(n) (25)

for

x < x0+ lim

n!1

n!

a_nB^m(n)

1=n:

3.2 Special Products and Sums Representing RZF

In this section, Theorem 1 with the intention of relating partial zeta function and partial primes’ product function (De…nition2) to RZF is presented. However, a simple average sum based on limit operator is also introduced in Theorem6.

A limit-based relation for _n is

(s) = lim

N!1

1 N

XN n=1

n(s); (26)

which is proved by a relation presented in Lemma4.

Lemma 4 Let S be a Cesàro mean [12] (sum or product) andS_n its partial value. Then

S= lim

N!1

1 N

XN n=1

S_n;

where

S_n= Xn m=1

a_n= Yn m=1

b_n;

under condition

S = lim

n!1Sn: Proof. The sum in statement of Lemma4is written as

S= lim

N!1

n S₁

|{z}N

0

+ S₂

|{z}N

0

+ +S_n

N +S_n+1

N + +S₁

| {z N}

average of inf inite terms

o

=S₁;

which ends proof under condition of lemma with assumption of

nlim!1

sn

n = 0:

De…nition 2 De…ne partial primes’ product function and partial zeta function as Pn(s) =

nYⁿ

i=1

pi

o ^s

=fpn#g ^s (27)

and

n(s) = Yn i=1

1

1 p_i^s; (28)

wherepn# is the primorial function.

Next, using these functions, Theorem5 is introduced.

Theorem 5 RZF can be determined in terms of partial product functions Pn and _n as

(s) = lim

n!1

1 P_2bnc+1(s)

n 2 ^s 1 2 ^s

2Xbnc k=1

( 1)^k+1P_k(s) _k+1(s)o

; R(s)>1 (29)

and

(s) = 2^s 2^s 1 +

X1 n=2

Pn(s) _n(s)

P_{n 1}(s) ; R(s)>1: (30)

Proof. By assumingp_i as sequence of prime numbers, left-side of equality p₁^s

1 p₁^s = p₁^s (1 p₁^s)(1 p₂^s)

(p1p2) ^s (1 p₁^s)(1 p₂^s)

is expanded as its right-side by multiplying numerator and denominator by 1 p₂^s. If this is done for n prime numbers, it yields

p₁^s

1 p₁^s = p₁^s (1 p₁^s)(1 p₂^s)

(p1p2) ^s

(1 p₁^s): : :(1 p₃^s)+ +( 1)ⁿ(p1: : : pn 1) ^s

(1 p₁^s): : :(1 pn^s)+( 1)ⁿ⁺¹(p1: : : pn) ^s (1 p₁^s): : :(1 pn^s): This can be rewritten on the basis of partial product functionsP_n and _n as

p₁^s

1 p₁^s =P₁(s) ₂(s) P₂(s) ₃(s) + + ( 1)ⁿP_{n 1}(s) _n(s) + ( 1)ⁿ⁺¹P_n(s) _n(s):

In this step,nis replaced by2n+ 1in order to eliminate last term sign, then equation is solved for _2n+1(s), therefore it gives

2n+1(s) = 1 P2n+1(s)

n p₁^s 1 p₁^s

X2n

k=1

( 1)^k+1Pk(s) _k+1(s) o

which equals Theorem5 statement by settingp1= 2and limiting equality asnapproaches in…nity.

To prove second relation, same technique by interchanging …rst and second terms is applied which results in

p₁^s

1 p₁^s = p₁^sp₂^s

(1 p₁^s)(1 p₂^s)+ p1 s

(1 p₁^s)(1 p₂^s): This is done fornprime numbers, therefore it gives

1

1 p₁^s+ + p_n ^s

(1 p₁^s): : :(1 pn^s)= 1

(1 p₁^s): : :(1 pn^s); which can be reformed as (30) byp_k =P_k(s)=P_{k 1}(s).

A familiar relation [13] fors= 2;3; : : : is (s) = 2^s

2^s 1+ X1 n=2

(p_{n 1}#)^s

Js(pn#); s= 2;3; : : :

in terms of primorial sequencep_n# with multiplicative Jordan’s totient function de…ned as p_n# =

Yn k=1

p_k

and

Jk(n) =n^kY

pjn

(1 1 p^k):

However, equation (29) is also valid for non-integer values ofsin the domain of convergence.

Theorem 6 The average identity

(s) = 1 q

nX^q

i=1

n 1 1 p_i^s +

X1 n=0

pⁿ⁺¹_iX1 m=pⁿ_i+1

1 m^s

oo

(31)

represents RZF, where sequencepi denotes prime numbers.

Proof. The set

[n i=0

n

mⁱ+ 1; mⁱ+ 2; : : : ; mⁱ⁺¹ 1 o

;

where1< m2N;is equal tof1; 2; : : : ; mⁿ⁺¹ 1gexcluded byf1; m¹; m²: : : ; mⁿ⁺¹ g, so the sum X1

n=0 pⁿ⁺¹_iX1 m=pⁿ_i+1

1

m^s = X

m2N f1; pi; p²_i; :::g

1 m^s

in (31) equals

(s) 1 + 1 p^s_i + 1

p^2s_i +: : : = (s) 1 1 p_i ^s; and it …nally follows that

(s) =1 q

nX^q

i=1

n 1

1 p_i^s + (s) 1 1 p_i ^s

oo : The last relation is theorem’s statement.

Equation (31) is valid for other unique sequences satisfying 1< i 2Nin place ofpi, but using prime numbersp_i with theorem 2 and relation (31), a functional equation for Euler product is found as

Y1 i=1

1 1 p_i ^s =

Y1 i=1

n (s)

X1 n=0

pⁿ⁺¹_iX1 m=pⁿ_i+1

1 m^s

o

; (32)

where terms in left-hand side correspond to terms in right-hand side respectively.

4 Summary

Basic de…nitions related to the RZF were reviewed brie‡y at beginning part of this paper. Then, several lemmas with their proofs and relations to the RZF were presented in order to develop some power series (with RZF in their coe¢ cients) and some limit-based identities. Finally, a theorem providing a familiar identity for RZF, and another one for Euler product were introduced.

Acknowledgment. The author appreciates reviewers for their suggestions which improved this work.

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