The recurrence formulas for primes and non-trivial zeros of the Riemann zeta function
aa r X i v : . [ m a t h . G M ] S e p The recurrence formulas for primes andnon-trivial zeros of the Riemann zeta function
Artur Kawalec
Abstract
In this article, we explore the Riemann zeta function with a perspec-tive on primes and non-trivial zeros. We develop the Golomb’s recurrenceformula for the n th+1 prime, and assuming (RH), we propose an analyti-cal recurrence formula for the n th+1 non-trivial zero of the Riemann zetafunction. Thus all non-trivial zeros up the n th order must be known togenerate the n th+1 non-trivial zero. We also explore a variation of therecurrence formulas for primes based on the prime zeta function, whichwill be a basis for the development of the recurrence formulas for thenon-trivial zeros based on the secondary zeta function. In the last part,we review the presented formulas and outline the duality between primesand non-trivial zeros. The proposed formula implies that all primes canbe converted into an individual non-trivial zero, and conversely, all non-trivial zeros can be converted into an individual prime. Also, throughoutthis article, we summarize numerical computation and verify the presentedresults to high precision. The Riemann zeta function is defined by the infinite series ζ ( s ) = ∞ X n =1 n s , (1)which is absolutely convergent for ℜ ( s ) >
1, where s = σ + it is a complexvariable. The values for the first few special cases are: ζ (1) ∼ k X n =1 n ∼ γ + log( k ) as k → ∞ ,ζ (2) = π ,ζ (3) = 1 . . . .,ζ (4) = π ,ζ (5) = 1 . . . .. (2)1or s = 1, the series diverges asymptotically as γ +log( k ), where γ = 0 . . . . is the Euler-Mascheroni constant. The special values for even positive integerargument are given by the Euler’s formula ζ (2 k ) = | B k | k )! (2 π ) k , (3)for which the value is expressed as a rational multiple of π k where the constants B k are Bernoulli numbers denoted such that B = 1, B = − / B = 1 / ζ ( s ) converge tounique constants, which are not known to be expressed as a rational multipleof π k +1 as occurs in the even positive integer case. For n = 3, the value iscommonly known as Ap´ery’s constant, who proved its irrationality.At the heart of the Riemann zeta function are prime numbers, which areencoded by the Euler’s product formula ζ ( s ) = ∞ Y n =1 (cid:18) − p sn (cid:19) − (4)also valid for ℜ ( s ) >
1, where p = 2, p = 3, and p = 5 and so on, denotethe prime number sequence. The expression for the complex magnitude, ormodulus, of the Euler prime product is | ζ ( σ + it ) | = ζ (4 σ ) ζ (2 σ ) ∞ Y n =1 (cid:18) − cos( t log p n )cosh( σ log p n ) (cid:19) − (5)for σ >
1, which for a positive integer argument σ = k simplifies the zeta termsusing (3), resulting in | ζ ( k + it ) | = (2 π ) k s | B k | (2 k )! | B k | (4 k )! ∞ Y n =1 (cid:18) − cos( t log p n )cosh( k log p n ) (cid:19) − / . (6)Using this form, the first few special values of this representation are ζ (1) ∼ π √ k Y n =1 (cid:18) − p n + p − n (cid:19) − / ∼ e γ log( p k ) ,ζ (2) = π √ ∞ Y n =1 (cid:18) − p n + p − n (cid:19) − / ,ζ (3) = π r ∞ Y n =1 (cid:18) − p n + p − n (cid:19) − / ,ζ (4) = π r ∞ Y n =1 (cid:18) − p n + p − n (cid:19) − / ,ζ (5) = π r ∞ Y n =1 (cid:18) − p n + p − n (cid:19) − / , (7)2here we let t = 0 and reduced the hyperbolic cosine term [6][8]. The value for ζ (1) in terms of Euler prime product representation is asymptotic to e γ log( p k )due to Mertens’s theorem as k → ∞ [4][13]. Also, the arg of the Euler productcan be found asarg ζ ( σ + it ) = − ∞ X n =1 tan − (cid:18) sin( t log p n ) p σn − cos( t log p n ) (cid:19) (8)thus writing the Euler product in polar form: ζ ( s ) = | ζ ( s ) | e i arg ζ ( s ) . (9)The Euler prime product permits the primes to be individually extracted fromthe infinite product under certain limiting conditions, as we have shown in [5],thus yielding the Golomb’s formula for primes [3]. To illustrate this, when weexpand the product we have ζ ( s ) = (cid:18) − p s (cid:19) − (cid:18) − p s (cid:19) − (cid:18) − p s (cid:19) − . . . , (10)and next, we wish to solve for the first prime p , then we have p = (cid:18) − ǫ ( s ) ζ ( s ) (cid:19) − /s , (11)where ǫ k ( s ) = ∞ Y n = k (cid:18) − p sn (cid:19) − (12)is the tail of Euler product starting at p k . When we then consider the limit p = lim s →∞ (cid:18) − ǫ ( s ) ζ ( s ) (cid:19) − /s , (13)then ǫ ( s ) → ζ ( s ) ∼ O ( p − s ), while ǫ ( s ) ∼ O ( p − s ), and the gap p − s ≫ p − s is only wideningas s → ∞ , hence the contribution due to Riemann zeta function dominates thelimit, and the formula for the first prime becomes p = lim s →∞ (cid:18) − ζ ( s ) (cid:19) − /s . (14)Numerical computation of (14) for s = 10 and s = 100 is summarized in Table1, and we observe convergence to p . The next prime in the sequence is foundthe same way by solving for p in (10) to obtain p = lim s →∞ − (cid:16) − p s (cid:17) − ǫ ( s ) ζ ( s ) − /s , (15)3here similarity as before, ǫ ( s ) → − p − s ) − as s → ∞ , where it cancels the first prime product in ζ ( s ), so that (1 − p − s ) ζ ( s ) ∼ O ( p − s ), while ǫ ( s ) ∼ O ( p − s ), and the gap p − s ≫ p − s is increasingrapidly as s → ∞ , hence the contribution due to Riemann zeta function andthe first prime product dominates the limit, and we have p = lim s →∞ − (cid:16) − p s (cid:17) − ζ ( s ) − /s . (16)Numerical computation of (16) for s = 10 and s = 100 is summarized in Table 1,and we observe convergence to p . And the next prime follows the same pattern(1 − p − s )(1 − p − s ) ζ ( s ) ∼ O ( p − s ), while ǫ ( s ) ∼ O ( p − s ) which results in p = lim s →∞ − (cid:16) − p s (cid:17) − (cid:16) − p s (cid:17) − ζ ( s ) − /s . (17)Hence, this process continues for the n th+1 prime, and so if we define a partialEuler product up to the n th order as Q n ( s ) = n Y k =1 (cid:18) − p sk (cid:19) − (18)for n > Q ( s ) = 1, then we obtain the Golomb’s formula for the p n +1 prime p n +1 = lim s →∞ (cid:18) − Q n ( s ) ζ ( s ) (cid:19) − /s . (19)We performed numerical computation of (19) in PARI/GP software package, asit is an excellent platform for performing arbitrary precision computations [9],and its functionality will be very useful for the rest of this article. Before runningany script, we recommend to allocate alot of memory allocatemem(1000000000) ,and setting precision to high value, for example \ p 2000 . We tabulate thecomputational results in Table 1 for s = 10 and s = 100 case, and observe theconvergence approaching to the p n +1 based on the knowledge of all primes upto the n th order. For the p case, s is still too small, hence we performed avery high precision computation for n = 9999 and s = 10000 with precision setto 50000 decimal places, and now the true value of the prime is revealed: p ≈ . . . .. (20)This formula will always converge because p − sn ≫ p − sn +1 as s → ∞ , and alsobecause the prime gaps are always bounded which will prevent higher orderprimes from modifying the main asymptote. It’s just a matter of allowing the4imit variable s to tend a large value, however, as it seen it is not very practicalfor computing large primes, as very high arbitrary precision is required. Thescript in PARI is shown in Listing 1 to compute the next prime using theGolomb’s formula (19), which was used to compute Table 1. The precisionmust be set very high, we generally set to 2000 digits by default.Table 1: The p n +1 prime computed by equation (19) shown to 15 decimal places. n p n +1 s = 10 s = 1000 p p p p p p p p p p p p \\ D e f i n e p a r t i a l E u l e r pr o duct up t o nth o r d e rQn( x , n)= { prod ( i =1 ,n ,(1 −
1/ prime ( i ) ˆ x ) ˆ ( − }\\ Compute t h e next prime { n=10; \\ s e t ns =100; \\ s e t l i m i t v a r i a b l e \\ compute next primepnext=(1 − Qn( s , n ) / z e t a ( s ))ˆ( −
1/ s ) ;p r i n t ( pnext ) ; } Listing 1: PARI script for computing equation (19).5he Riemann zeta function has many representations. One common form isthe alternating series representation ζ ( s ) = 11 − − s ∞ X n =1 ( − n +1 n s , (21)which is convergent for ℜ ( s ) >
0, with some exceptions at ℜ ( s ) = 1 due theconstant factor. By application of the Euler-Maclaurin summation formula, themain series (1) can also be extended to domain ℜ ( s ) > ζ ( s ) = lim k →∞ k − X n =1 n s − k − s − s . (22)Equations (21) and (22) are hence valid in the critical strip region 0 < ℜ ( s ) < s = 1 thatgives a globally convergent series valid anywhere in the complex plane exceptat s = 1 as ζ ( s ) = 1 s − ∞ X n =0 ( − n γ n ( s − n n ! . (23)The coefficients γ n are the Stieltjes constants, and γ = γ is the usual Euler-Mascheroni constant. We observe that γ n are linear in the series, hence ifwe form a system of linear equations, then using the Cramer’s rule and someproperties of an Vandermonde matrix, we find that Stieltjes constants can berepresented by determinant of a certain matrix: γ n = ± det( A n +1 ) (24)where the matrix A n ( k ) is matrix A ( k ), but with an n th column swapped witha vector B as given next A ( k ) = −
11! 1 − . . . k k ! −
21! 2 − . . . k k ! −
31! 3 − . . . k k ! ... ... ... ... . . . ...1 − ( k +1)1! ( k +1) − ( k +1) . . . ( k +1) k k ! (25)and B ( k ) = ζ (2) − ζ (3) − ζ (4) − ... ζ ( k + 1) − k . (26)6he ± sign depends on k , but to ensure a positive sign, the size of k must be amultiple of 4. Hence, the first few Stieltjes constants can be represented as: γ = lim k →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ (2) − −
11! 1 − . . . k k ! ζ (3) − −
21! 2 − . . . k k ! ζ (4) − −
31! 3 − . . . k k ! ... ... ... ... . . . ... ζ ( k + 1) − k − ( k +1)1! ( k +1) − ( k +1) . . . ( k +1) k k ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (27)and the next Stieltjes constant is γ = lim k →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ (2) − − . . . k k ! ζ (3) −
12 2 − . . . k k ! ζ (4) −
13 3 − . . . k k ! ... ... ... ... . . . ...1 ζ ( k + 1) − k ( k +1) − ( k +1) . . . ( k +1) k k ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (28)and the next is γ = lim k →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ζ (2) − − . . . k k ! − ζ (3) − − . . . k k ! − ζ (4) − − . . . k k ! ... ... ... ... . . . ...1 − ( k +1)1! ζ ( k + 1) − k − ( k +1) . . . ( k +1) k k ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (29)and so on. In Table 2, we compute the determinant formula (24) for the first 10Stieltjes constants for k = 500, and observe the convergence. In Listing 2, thescript in PARI to generate values for Table 2 is also given. This shows that the γ n constants can be represented by ζ ( n ) at positive integer values as basis γ n = lim k →∞ ( C n, ( k ) + k +1 X m =2 ( − m C n,m ( k ) ζ ( m ) ) (30)where the expansion coefficients C n,m are rational and divergent, which growvery fast as k increases. The index n ≥ n th Stieltjes constant, andindex m ≥ ζ ( m ) basis value. These coefficients can be generatedby expanding the determinant of A n using the Leibniz determinant rule alongcolumns with the zeta values. For example, for k = 12, which is a multiple of4, then the first few expansion coefficients are γ ≈ − ζ (2) − ζ (3) + 220 ζ (4) − ζ (5) + 792 ζ (6) − . . . (31)7he C , coefficient is the harmonic number H C , = − H k = − k X n =1 n (32)and the next are C ,m = (cid:18) km − (cid:19) . (33)For the next γ n , the first few coefficients for k = 12 are γ ≈ − ζ (2) − ζ (3)+ 76781126 ζ (4) − ζ (5)+ 8047735 ζ (6) − . . . , (34)and for the next γ n , we have γ ≈ − ζ (2) − ζ (3)+ 143644105 ζ (4) − ζ (5)+ 574108105 ζ (6) − . . . , (35)and so on, but these coefficients are more difficult to determine and they divergevery fast.Table 2: The first 30 digits of γ n computed by equation (24) for k = 500. n γ n Significant Digits0 0.577215664901532860606512090082 341 -0.072815845483676724860586375874 342 -0.009690363192872318484530386035 333 0.002053834420303345866160046542 324 0.002325370065467300057468170177 315 0.000793323817301062701753334877 306 -0.000238769345430199609872421842 297 -0.000527289567057751046074097507 298 -0.000352123353803039509602052177 289 -0.000034394774418088048177914691 2810 0.000205332814909064794683721922 268 n = 0 ; \\ s e t nth S t i e l t j e s c o n s t a n tk = 1 0 0 ; \\ s e t l i m i t v a r i a b l eAn=ma tr ix ( k , k ) ; \\ a l l o c a t e ma tr ix \\ l o a d ma tr ix An for ( j =1 ,k , for ( i =1 ,k , i f ( j==1+n , An [ i , j ]= z e t a ( i +1) − − i ) ˆ ( j − − \\ compute d e t e r m i n a n t o f Anyn = matdet (An ) ;p r i n t ( yn ) ; } Listing 2: PARI script for computing equation (24)The Hadamard infinite product formula is another global analytically con-tinued representation of (1) to the whole complex plane ζ ( s ) = π s/ s − s/ ∞ Y n =1 (cid:18) − sρ n (cid:19) (36)having a simple pole at s = 1, and at the heart of this form is an infinity ofcomplex non-trivial zeros ρ n = σ n + it n , which are constrained to lie in thecritical strip 0 < ℜ ( s ) < σ = 1 /
2. It is not yet known whether thereare non-trivial zeros off of the critical line in the range 0 < ℜ ( s ) < σ = 1 /
2, a problem of the Riemann Hypothesis (RH). To date, there has beena very large number of zeros verified numerically to lie on the critical line, andnone was ever found off of the critical line. The first few non-trivial zeros onthe critical line ρ n = 1 / it n have imaginary components t = 14 . ... , t = 21 . ... , t = 25 . ... which were originally found numericallyusing a solver, but if (RH) is true, then can be computed analytically. Also, wewill interchangeably refer to ρ n or t n to imply a non-trivial zero.The Hadamard product representation can be interpreted as a volume ofan s-ball (that is for a ball of complex dimension s ). For a positive integer n ,the n-ball defines all points satisfying Ω = { x + x + x · · · + x n ≤ R n } , andintegrating gives the total volume V ( n ) = Z Z Z . . . Z x + x + x ... + x n ≤ R n dx dx dx . . . dx n = K ( n ) R n , (37)9here K ( n ) = π n/ Γ(1 + n/
2) (38)is the proportionality constant. Now, generalizing the n-ball to an s-ball ofcomplex s dimension for ζ ( s ), we can identify that the terms involving π andΓ( s ) function is K ( s ), and that the radius of the s-ball is the remaining productinvolving the non-trivial zeros R ( s ) s = 12( s − ∞ Y n =1 (cid:18) − sρ n (cid:19) , (39)which is actually the Riemann xi function ξ ( s ) multiplied by 1 / ( s − ζ ( s ) = V s = K ( s ) R ( s ) s (40)can be understood as a volume quantity, which when packed into an s-ball, thenthe radius function in this form is being described by explicitly the non-trivialzeros. The trivial zeros at negative even integers − , − , − . . . − n are thenthe zeros of the proportionality constant due to the pole of Γ( s ). For example,if we consider s = 2, then ζ (2) = K (2) R (2) = πR (41)where R = p π/ . . . . is the radius to give the volume quantityfor ζ (2), which from (1) can be understood as packing the areas of squares with1 /n sides into a circle. And similarly for s = 3 ζ (3) = K (3) R (3) = 43 πR (42)where R = 0 . . . . is the radius to give the volume quantity for Ap´ery’sconstant ζ (3), which from (1) can be understood as packing the volumes ofcubes with 1 /n sides into a sphere. Hence in this view, the non-trivial zerosare governing the radius quantity of an s-ball, essentially encoding the volumeinformation of ζ ( s ), and while the trivial zeros are just the zeros of the pro-portionality constant K ( s ), which has a role of scaling the values of non-trivialzeros across the dimension s to the values that they currently are, and perhapseven on the critical line. If we plot the radius in the range 1 < σ < ∞ , we find aminima for R which occurs between s = 2 and s = 3 at s min = 2 . ... and R min = 0 . . . . . That would mean that the s-ball would reachminimum radius R min at s min .If we consider the complex magnitude for ζ ( s ) for representations (21) and(22), and note that at each non-trivial zero on the critical line, a harmonic series10s induced from which we can obtain formulas for the Euler-Mascheroni constant γ expressed as a function of a single non-trivial on the critical line zero as γ = lim k →∞ ( k X v =1 k X u = v +1 ( − u ( − v +1 √ uv cos( t n log( u/v )) − log( k ) ) (43)and the second formula as γ = lim k →∞ ( ) + t n + 2 k − X v =1 k − X u = v +1 √ uv cos( t n log( u/v )) − log( k ) ) , (44)where it is assumed the index variables satisfy u > v starting with v = 1 [7][8].Thus, any individual non-trivial zero on the critical line t n can be converted to γ , which is independent on (RH). As a numerical example, for t and k = 10 ,we obtain γ = 0 . . . . accurate to 5 decimal places, however, thecomputation becomes more difficult as it grows as O ( k ) due to the doubleseries. And if we subtract equations (21) and (22), then we obtain a relation1 | ρ n | = 1( ) + t n = lim k →∞ √ k k X m =1 √ m cos( t n log( m/k )) (45)whereby any individual non-trivial zero can be converted to its absolute valueon the critical line. Also next, the infinite sum over non-trivial zeros ∞ X n =1 | ρ n | = 12 γ + 1 −
12 log(4 π ) , (46)is an example of secondary zeta function family.There is also another whole side to the theory of the Riemann zeta functionconcerning the prime counting function π ( n ) up to a given quantity n , and thenon-trivial zero counting function N ( T ) up to a given quantity T. It is naturalto take the logarithm of the Euler prime product yielding a sumlog[ ζ ( s )] = ∞ X n =1 ∞ X m =1 m p msn (47)from which motivates to define a function J ( x ) that increases by 1 at eachprime, by at prime square, by at prime cubes, and so on [2, p.22] and [14].Riemann then expressed J ( x ) by Fourier inversion as J ( x ) = 12 πi Z a + i ∞ a − i ∞ log[ ζ ( s )] x s s ds ( a > . (48)After finding a suitable expansion for log[ ζ ( s )] in terms of zeros as ξ ( s ) = 12 Y ρ (cid:18) − sρ (cid:19) , (49)11hen after a very detailed and lengthy analysis [2], the main formula for J ( x )appears as J ( x ) = Li( x ) − ∞ X n =1 Li( x ρ n ) − log(2) + Z ∞ x dtt ( t −
1) log( t ) (50)for x >
1, and then by applying M¨obius inversion, leads to recovering π ( x ) = ∞ X n =1 µ ( n ) n J ( x /n ) . (51)Hence, through this formula, the non-trivial zeros are shown to be involved inthe generation of primes. Although applying M¨obius inversion to recover π ( n )is somewhat circular in this case, because one needs to have knowledge of allthe primes by µ ( n ), but the main prime content is still in J ( x ), which comesfrom the contribution of non-trivial zero terms.Furthermore, in analysis by LeClair [10] concerning N ( T ), it is found that n th non-trivial zeros satisfy the following transcendental equation: t n π log (cid:18) t n πe (cid:19) + lim δ → π arg ζ ( 12 + it n + δ ) = n − , (52)however, the contribution to due to arg function is very small, and only providesfine level tuning to the overall equation, hence when dropping the arg term,LeClair obtained an approximate asymptotic formula for non-trivial zeros viathe Lambert function W ( x ) e W ( x ) = x transformation: t n ≈ π n − W (cid:16) n − e (cid:17) . (53)It turns out that this approximation works very well with an accuracy down toa decimal place. For example, with this formula, we can quickly approximate a10 zero: t ≈ . t n get better for higher zeros as n → ∞ . Infact, LeClair computed the largest non-trivial zero known to date for n = 10 using this method.Also, very little is known about the properties of non-trivial zeros. Forexample, they are strongly believed to be simple, but remains unproven. Andin the works by Wolf [15], a large sample of non-trivial zeros was numericallyexpanded into continued fractions, from which it was possible to compute theKhinchin’s constant, which strongly suggests they are irrational.12n this article, we propose an analytical recurrence formula for t n +1 , verysimilar to the Golomb’s formula for primes, thus all non-trivial zeros up to t n must be known in order to compute the t n +1 zero. The formula is based on acertain representation of the secondary zeta function Z ( s ) = ∞ X n =1 t sn (55)in the works of Voros [12], for s >
1, which is not involving non-trivial zeros, thusavoiding circular reasoning. There is alot of work already on the secondary zetafunctions published in the literature, especially concerning the meromorphicextension of Z ( s ) via the Mellin transform techniques and tools of spectraltheory.We now introduce the main result of this paper. Assuming (RH), the fullrecurrence formula for the t n +1 non-trivial zero is: t n +1 = lim m →∞ " ( − m +1 m − | ζ | ) (2 m ) (cid:0) (cid:1) + ∞ X k =1 (cid:0) + 2 k (cid:1) m − m ! − n X k =1 t mk − m (56)for n ≥
0, thus all non-trivial zeros up the n th order must be known in order togenerate the n th+1 non-trivial zero. This formula is a solution to ζ ( s ) = 0 (57)where s = ρ n = 1 / it n for σ n = 1 /
2, and the zeros t n are real and ordered t < t < t < . . . t n . This formula is satisfied by all representations of ζ ( s ) onthe critical strip, such as by (21), (22), (23), (36), and so on. And in the nextsections, we will develop this formula, and explore some its variations, and thenwe will numerically compute non-trivial zeros to high precision. We will alsodiscuss some possible limitations to this formula for n → ∞ .In the last section, we will discuss formulas for t n which actually can berelated to the primes themselves, and that one could compute t n as a functionof all primes. And conversely, one could compute any individual prime p n as afunction of all non-trivial zeros. The only open problem is whether all primescan be converted to all non-trivial zeros. n th+1 prime formula Golomb described several variations of the prime formulas of the form (19), onesuch is p n +1 = lim s →∞ [ ζ ( s ) − Q n ( s )] − /s , (58)which will serve to motivate the next result, which is based on the prime zetafunction, and that will then serve as a basis for the development of an analogueformula for the n th+1 non-trivial zero formula in the next section.13he prime zeta function is an analogue of (1), but instead of summing overreciprocal integer powers, we sum over reciprocal prime powers as P ( s ) = ∞ X n =1 p sn . (59)When we consider the expanded sum P ( s ) = 1 p s + 1 p s + 1 p s + . . . (60)then similarly as before, we wish to solve for p , and obtain1 p s = P ( s ) − p s − p s − . . . (61)which leads to p = (cid:18) P ( s ) − p s − p s − . . . (cid:19) − /s . (62)If we then consider the limit, p = lim s →∞ (cid:18) P ( s ) − p s − p s − . . . (cid:19) − /s (63)then we find that the higher order primes decay faster than P ( s ), namely, P ( s ) ∼ p − s , while the tailing error is O ( p − s ), and so P ( s ) dominates the limit. Since p − s ≫ p − s , hence we have p = lim s →∞ [ P ( s )] − /s . (64)To find p we consider (60) again p = lim s →∞ (cid:20) P ( s ) − p s − p s . . . (cid:21) − /s , (65)and when taking the limit, then we must keep p , while the higher order primesdecay faster, namely, P ( s ) − p − s ∼ p − s , while the tailing error is O ( p − s ), andso P ( s ) − p − s dominates the limit. Since p − s ≫ p − s , hence we have p = lim s →∞ (cid:20) P ( s ) − p s (cid:21) − /s . (66)And similarly, the next prime is found the same way, but this time we mustretain the two previous primes p = lim s →∞ (cid:20) P ( s ) − p s − p s (cid:21) − /s . (67)14ence in general, if we define a partial prime zeta function up to the n th order P n ( s ) = n X k =1 p sk , (68)then the n th+1 prime is p n +1 = lim s →∞ [ P ( s ) − P n ( s )] − /s . (69)At this point, knowing P ( s ) by the original definition (59) leads to circularreasoning, hence we seek to find other representations for P ( s ) that don’t involveprimes directly. We first explore the relationlog[ ζ ( s )] = ∞ X k =1 P ( ks ) k (70)and then by applying M¨obius inversion leads to P ( s ) = ∞ X k =1 µ ( k ) log( ks ) k , (71)where µ ( k ) is the M¨obius function, which however, still depends on the primes,so it may not be a good candidate for P ( s ). There is another equation for P ( s )in [11] using the recurrence relation P ( s ) = 1 − s ζ ( s ) − P (2 s ) (72)which leads to a nested radical representation P ( s ) = 1 − vuuuut ζ (2 s ) − vuuut ζ (2 s ) − vuut ζ (4 s ) − s ζ (8 s ) . . ., (73)that only depends on ζ ( s ), which could be computed by other means, such asby equation (1). It turns out that this nested radical formula is very slow toconverge, making it almost impractical to compute for s → ∞ . And if there areother representations for P ( s ) not involving primes, then one could certainlyuse them, but we are unaware of such.To verify this equation, we pre-compute P ( s ) using primes to high precision,thus introducing circular reasoning, since it is impractical to use an independentrepresentation (73). Hence, we pre-compute P ( s ) for s = 10 and s = 100 as P (10) = 9 . × − . . . (74)and P (100) = 7 . × − . . . . (75)15ext, we summarize computation for p n +1 by formula (69) in Table 3, andobserve the convergence to the p n +1 prime, just as the Golomb’s formula forprimes. For equation (69), we would also like to seek other representationfor P ( s ) not involving primes directly. And as before, the convergence worksbecause O ( p − sn ) ≫ O ( p − sn +1 ) as s → ∞ , (76)and also that the prime gaps are bounded, which prevents any higher orderprimes from modifying the main asymptote. Now we proceed to the next section.Table 3: The p n +1 prime computed by equation (69) shown to 15 decimal places. n p n +1 s = 10 s = 1000 p p p p p p p p p p The recurrence formula for non-trivial zeros
The secondary zeta function has been studied in the literature, and there hasbeen interesting developments concerning the analytical extension to the wholecomplex plane for Z ( s ) = ∞ X n =1 t sn (77)which has many parallels with the zeta function. In this article, the symbol Z is implied, and is not related to the Hardy-Z function. For the first few specialvalues the Z ( s ) yields Z (2) = 12 (log | ζ | ) (2) (cid:0) (cid:1) + 18 π + β (2) −
4= 0 . . . .,Z (3) = 0 . . . . ,Z (4) = −
112 (log | ζ | ) (4) (cid:0) (cid:1) − π − β (4) + 16= 3 . × − . . . ,Z (5) = 2 . × − . . . . (78)The special values for even positive integer argument Z (2 m ) is: Z (2 m ) = ( − m (cid:20) − m − | ζ | ) (2 m ) (cid:0) (cid:1) + − (cid:2) (2 m − ζ (2 m ) + 2 m β (2 m ) (cid:3) + 2 m (cid:21) (79)and is found in [12,p. 693] by works of Voros, and it’s originally denoted as Z (2 σ ). This formula is a sort of an analogue for Euler’s formula (3) for ζ (2 n ),and is valid for m ≥
1, where m is an integer, and β ( s ) is the Dirichlet betafunction β ( s ) = ∞ X n =0 ( − n (2 n + 1) s = ∞ Y n =1 (cid:18) − χ ( p n ) p sn (cid:19) − , (80)where χ is the Dirichlet character modulo 4. The value for β (2) is the Catalan’sconstant. In (78), the odd values for Z (2 m + 1) were computed numericallyby summing 25000 zeros, as it is not known whether there is a closed-formrepresentation similarly as for the ζ (2 m + 1) case, and so the given values couldonly be accurate to several decimal places. The formula (79) assumes (RH),and is a result of a complicated development to meromophically extend (77) to17he whole complex plane using tools from spectral theory. Furthermore, usingthe relation, also found in [12,p. 681] as12 s ζ (cid:0) s, (cid:1) = ∞ X k =1 (cid:0) + 2 k (cid:1) s = 2 s (cid:20) (cid:0) (1 − − s ) ζ ( s ) + β ( s ) (cid:1) − (cid:21) , (81)from which we have several variations of (79) for Z (2 m ) as Z (2 m ) = ( − m +1 " m − | ζ | ) (2 m ) (cid:0) (cid:1) + ∞ X k =1 (cid:0) + 2 k (cid:1) m − m (82)and another as Z (2 m ) = ( − m +1 (cid:20) m − | ζ | ) (2 m ) (cid:0) (cid:1) + 12 m ζ (cid:0) m, (cid:1) − m (cid:21) . (83)The expressions involving the log( | ζ | ) (2 m ) (cid:0) (cid:1) term can be computed numericallyand independently of the non-trivial zeros, and there is no known closed-formrepresentation of it, but there is for the odd valueslog( | ζ | ) (2 m +1) (cid:0) (cid:1) = 12 (2 m )!(2 m +1 − ζ (2 m + 1) + 14 π m +1 | E m | , (84)where E m are Euler numbers [12,p. 686]. Unfortunately, the log( | ζ | ) (2 m +1) ( )term is not involved in the computation of Z ( m ) for m >
1. Also, the infiniteseries in (81) is related to the Hurwitz zeta function, and it can also be separatedinto two parts involving the zeta function and the beta function, which can thenbe related to primes via the Euler product, which we will come back to shortly.Now we will follow the same program that we did for the prime zeta functionas outlined in equations (59) to (69). If we begin with the secondary zetafunction Z ( s ) = 1 t s + 1 t s + 1 t s + . . . (85)and then solving for t we obtain1 t s = Z ( s ) − t s − t s − . . . (86)and then we get t = (cid:18) Z ( s ) − t s − t s − . . . (cid:19) − /s . (87)If we then consider the limit t = lim s →∞ (cid:18) Z ( s ) − t s − t s − . . . (cid:19) − /s (88)18hen, since O ( Z ( s )) ∼ O ( t − s ), and so the higher order non-trivial zeros decayas O ( t − s ) faster than Z ( s ), and so Z ( s ) dominates the limit, hence we have t = lim s →∞ [ Z ( s )] − /s . (89)Now, substituting representation (82) for Z ( s ) into (89), and s is now assumedbe an integer as a limit variable 2 m , then we get a direct formula for t as t = lim m →∞ " ( − m +1 m − | ζ | ) (2 m ) (cid:0) (cid:1) + ∞ X k =1 (cid:0) + 2 k (cid:1) m − m ! − m . (90)Next we numerically verify this formula in PARI, and the script is shown inListing 3. We broke up the representation (83) into several parts A to D. Also,sufficient memory must be allocated and precision set to high before running thescript. We utilize the Hurwitz zeta function representation, since it is availablein PARI, and the derivnum function for computing the m th derivative veryaccurately for high m . The results are summarized in Table 4 for various limitvalues of m from low to high, and we can observe the convergence to the realvalue as m increases. Already at m = 10 we get several digits of t , and at m =100 we get over 30 digits. We performed even higher precision computations,and the result is clearly converging to t .Table 4: The computation of t by equation (90) for different m .m t (First 30 Digits) Significant Digits1 6.578805783608427637281793074245 02 12.806907343833847091925940068962 03 13.809741306055624728153992726341 04 14.038096225961619450676758199577 05 14.102624784431488524304946186056 16 14.123297656314161936112154413740 17 14.130464459254236820197453483721 28 14.133083993992268169646789606564 29 14.134077755601528384660110026302 310 14.134465134057435907124435534843 315 14.134721950874675119831881762569 520 14.134725096741738055664458081219 625 14.134725141055464326339414131271 950 14.134725141734693789641535771021 16100 14.134725141734693790457251983562 3419 \\ s e t l i m i t v a r i a b l em = 2 5 0 ; \\ compute p a r a m e t e r s A t o DA = derivnum ( x=1/2 , l o g ( z e t a ( x ) ) , 2 ∗ m) ;B = 1 / ( f a c t o r i a l ( 2 ∗ m − ∗ m) ;D = (2ˆ( − ∗ m) ) ∗ z e t a h u r w i t z ( 2 ∗ m, 5 / 4 ) ; \\ compute Z( 2m)Z = ( − ∗ ( 1 / 2 ) ∗ (A ∗ B − C+D) ; \\ compute t 1t 1 = Zˆ( − ∗ m) ) ;p r i n t ( t 1 ) ; } Listing 3: PARI script for computing equation (90).Next, we perform a higher precision computation for m = 250 case, and theresult is t = 14 . . . . (91)accurate to 87 decimal places. In order to find the second non-trivial zero, wecomeback to (85), and solving for t yields t = lim s →∞ (cid:18) Z ( s ) − t s − t s − . . . (cid:19) − /s (92)and since the higher order zeros decay faster than Z ( s ) − t − s , we then have t = lim s →∞ (cid:18) Z ( s ) − t s (cid:19) − /s (93)and the zero becomes t = lim m →∞ " ( − m +1 m − | ζ | ) (2 m ) (cid:0) (cid:1) + ∞ X k =1 (cid:0) + 2 k (cid:1) m − m ! − t m − m . (94)A numerical computation for m = 250 yields t = 21 . . . . (95)20hich is accurate to 38 decimal places, and we assumed t used was alreadypre-computed to 2000 decimal places by other means. We cannot use the same t computed earlier with same precision, as it will cause self-cancelation in(85), and so the accuracy of t n must be much higher than t n +1 to guaranteeconvergence. And continuing on, the next zero is computed as t = lim m →∞ " ( − m +1 m − | ζ | ) (2 m ) (cid:0) (cid:1) + ∞ X k =1 (cid:0) + 2 k (cid:1) m − m ! − t m − t m − m . (96)A numerical computation for m = 250 yields t = 25 . . . . (97)which is accurate to 43 decimal places, and we assumed t and t was used tohigh enough precision which was 2000 decimal places in this example. Hence,just like for the n th+1 Golomb prime recurrence formulas and the prime zetafunction P ( s ), the same limit works for non-trivial zeros. As a result, if wedefine a partial secondary zeta function up to the n th order Z n ( s ) = n X k =1 t sk , (98)then the n th+1 non-trivial zero is t n +1 = lim m →∞ [ Z ( m ) − Z n ( m )] − /m (99)and the main formula: t n +1 = lim m →∞ " ( − m +1 m − | ζ | ) (2 m ) (cid:0) (cid:1) + ∞ X k =1 (cid:0) + 2 k (cid:1) m − m ! − n X k =1 t mk − m . (100)One can actually use any number of representations for Z ( s ), and the challengewill be find more efficient algorithms to compute them. And finally, we reporta numerical result for Z (500) as: Z = 7 . × − . . . . (101)From this number, we extracted the first 10 non-trivial zeros, which are summa-rized in Table 5 for k = 250. The previous non-trivial zeros used were alreadyknown to high precision to 2000 decimal places in order to compute the t n +1 .One cannot use the same t n obtained earlier because it will cause self-cancelation21n (85), and the accuracy for t n must be much higher than t n +1 to ensure con-vergence. Initially we started with an accuracy of 87 digits after decimal placefor t , and then it dropped to 7 to 12 digits by the time it gets to t zero.There is also a sudden drop in accuracy when the gaps get too small. And asthe gap gets too small, it has to be overcompensated by increasing m higher.Hence, these formulas are not very practical for computing high zeros as largenumerical precision is required, especially when we get to the first Lehmer pairat t = 7005 . ∼ .
04. Also, theaverage gap between zeros gets smaller as t n +1 − t n ∼ π log( n ) , making the use ofthis formula progressively harder and harder to compute. The open problem iswhether all non-trivial zeros can be generated in this fashion.Table 5: The t n +1 computed by equation (100). n t n +1 m = 250 Significant Digits0 t t t t t t t t t t We outline the duality between primes and non-trivial zeros. The Golomb’srecurrence formula (19) is an exact formula for the n th+1 prime p n +1 = lim s →∞ (cid:18) − Q n ( s ) ζ ( s ) (cid:19) − /s , (102)and the Hadarmad product formula establishes ζ ( s ) as a function of non-trivialzeros: ζ ( s ) = π s/ s − s/ ∞ Y n =1 (cid:18) − sρ (cid:19) . (103)Hence, this is a pathway from non-trivial zeros to the primes and without as-suming (RH), as the Hadamard product is over all zeros. On the other hand,22ecurrence formula for the n th+1 non-trivial zero is t n +1 = lim m →∞ " ( − m +1 " m − | ζ | ) (2 m ) (cid:0) (cid:1) − m +1 ++ 2 m − h (1 − − m ) ζ (2 m ) + β (2 m ) i − n X k =1 t k − m (104)where now one could substitute the Euler product for the zeta and beta func-tions, or both, which is what we will do next. We have (cid:0) − − m (cid:1) ζ (2 m ) = ∞ Y n =2 (cid:18) − p mn (cid:19) − (105)and β (2 m ) = ∞ Y n =2 (cid:18) − χ ( p n ) p mn (cid:19) − . (106)As a result, t n +1 = lim m →∞ " ( − m +1 " m − | ζ | ) (2 m ) (cid:0) (cid:1) − m +1 ++ 2 m − h ∞ Y n =2 (cid:0) − p − mn (cid:1) − + ∞ Y n =2 (cid:0) − χ ( p n ) p − mn (cid:1) − i − n X k =1 t k − m (107)which completes the pathway from primes to non-trivial zeros. We note thatthese formulas are independent, and thus avoid any circularity, however, therecurrence formula for t n +1 is dependent on (RH). And finally, in Appendix A,we present a PARI script to compute (107) recursively for several zeros. We explored various representations of the Riemann zeta function, such as theEuler prime product, the Laurent expansion, and the Golomb’s recurrence for-mula for primes. The Golomb’s formula is a basis for developing similar re-currence formulas for the n th+1 non-trivial zeros via an independent formulafor the secondary zeta function Z (2 m ), which does not involve non-trivial ze-ros. Hence, the non-trivial zeros can be extracted under the right excitationin the limit, just like prime numbers. We verified these formulas numerically,and they indeed do converge to t n +1 . The difficultly lies in computation of thelog( | ζ | ) (2 m ) ( ) term. We utilized the PARI/GP software package for comput-ing Z (2 m ) for m = 250, and the first zero t achieves 87 correct digits after thedecimal place. Presently, computing beyond that caused the test computer to23un out of memory. And so, if better and more efficient methods for comput-ing Z (2 m ) are developed, then more higher zeros can be computed accurately.But even then, computing up to a millionth zero for example, would be almostinsurmountable. The only open question is whether the recurrence for the non-trivial zeros will hold up, namely the limit O ( t − sn ) ≫ O ( t − sn +1 ) as s → ∞ , as theaverage gap between non-trivial zeroes decreases t n +1 − t n ∼ π log( n ) as n → ∞ .In case of the Golomb’s formula for primes, this gap is bounded.These formulas also suggest a new criterion for (RH). It suffices to take afirst zero t represented by (85) which depends on (RH) as t = lim m →∞ " ( − m +1 m − | ζ | ) (2 m ) (cid:0) (cid:1) + ∞ X k =1 (cid:0) + 2 k (cid:1) m − m ! − m (108)and passing it through to any number of representations of ζ ( s ) valid in thecritical strip to work out ζ ( 12 + it ) = 0 . (109)For example, if we take equation (45) and substitute t as1 | ρ | = 1( ) + t = lim k →∞ √ k k X m =1 √ m cos( t log( m/k )) , (110)then recovering t would imply (RH) if there was a way work out the series.Also, given that the recurrence formula is also an analogue of the prime zetafunction formula (59), we wonder whether there is another formula for P ( s ) asa function of non-trivial zeros involving terms of Z ( s ). And secondly, we alsowould like to see a formula for the secondary beta function B ( s ) = ∞ X n =1 r sn , (111)where r n are imaginary components of non-trivial zeros of β ( s ). For exam-ple, the first few zeros are r = 6 . ... , r = 10 . ... , r =12 . ... . The formula for B (2 m + 1) would probably have a term likelog( | β | ) (2 m +1) ( ). Then, the proposed recurrence formula would be r n +1 = lim s →∞ [ B ( s ) − B n ( s )] − /s , (112)where B n ( s ) = n X n =1 r sn (113)is the partial secondary beta function up to the n th order. And just like for theDirichlet beta, the same could potentially apply to other Dirichlet L-functions.Finally, we highlighted the duality between primes and non-trivial zeroswhere it is possible convert non-trivial zeros into an individual prime, and con-versely, to convert all primes into an individual non-trivial zero. But it is openproblem whether it is possible to convert all primes into all non-trivial zeros.24 eferences [1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functionswith Formulas, Graphs, and Mathematical Tables . Dover Publications, ninthprinting, New York, (1964).[2] H.M. Edwards.
Riemann’s Zeta Function . Dover Publications, Mineola, NewYork (1974).[3] S. Golomb.
Formulas for the next prime . Pacific Journal of Mathematics, ( An Introduction to the Theory of Numbers .Oxford Science Publications, (1980).[5] A. Kawalec.
The n th+1 prime limit formulas .math.GM/arXiv:1608.01671v2,(Aug. 2016).[6] A. Kawalec. Prime product formulas for the Riemann zeta function andrelated identities . math.GM/1901.09519v4, (Oct. 2019).[7] A. Kawalec.
Asymptotic formulas for harmonic series in terms of a non-trivial zero on the critical line . Computational Methods in Science and Tech-nology, (4), 161–166, (2019).[8] A. Kawalec. On the complex magnitude of Dirichlet beta function . Compu-tational Methods in Science and Technology, (1), 21–28, (2020).[9] The PARI Group, PARI/GP version , Univ. Bordeaux, (2019).[10] A. LeClair, An Electrostatic Depiction of the Validity of the Riemann Hy-pothesis and a Formula for the N-th Zero at Large N , International Journalof Modern Physics A ,1350151 (2013).[11] M. Vassilev-Missana. A note on prime zeta function and Riemann zetafunction . Notes on Number Theory and Discrete Mathematics, (4), 12-15,(2016).[12] A. Voros. Zeta functions for the Riemann zeros . Ann. Institute Fourier, ,665–699,(2003).[13] M. Wolf. .math.NT/1904.09855, (Apr. 2019).[14] M. Wolf. Will a physicist prove the Riemann hypothesis? . Reports onProgress in Physics, (3), 036001,(2020).[15] M. Wolf. Two Arguments that the Nontrivial Zeros of the Riemann ZetaFunction are Irrational. Part II . Computational Methods in Science andTechnology, (2), 47–53, (2020). Email: [email protected] Appendix A
The script in Listing 4 computes the n th+1 non-trivial from a set of primes byequation (107). The parameter pmax specifies the number of primes to use forthe Euler product. The starting limiting variable is m , and at each iteration m is decreased by a pre-set amount step m , so that the accuracy for t n will begreater than for t n +1 in order to avoid self-cancelation. The values for computedzeros are stored in an array, and the partial secondary zeta Z n is computed at { m = 2 5 0 ; \\ s t a r t i n g l i m i t v a r i a b l e mstep m = − \\ d e c r e a s e l i m i t step mpmax = 2 0 0 0 ; \\ s e t max number o f pr imes t o usetn = v e c t o r ( 1 0 0 ) ; \\ a l l o c a t e v e c t o r t o ho ld z e r o sn=1; \\ i n i t non − t r i v i a l z e r o c o u n t e r \\ s t a r t l o o p while (m != 0 , \\ compute p a r a m e t e r s A t o DA = derivnum ( x=1/2 , l o g ( z e t a ( x ) ) , 2 ∗ m) ;B = 1 / ( f a c t o r i a l ( 2 ∗ m − ∗ m+ 1 ); D = 2 ˆ (2 ∗ m − \\ compute E u l e r p r o d u c t sP1 = prod ( i =2 ,pmax,(1 −
1/ prime ( i ) ˆ ( 2 ∗ m) ) ˆ ( − − ( − − ∗ m) ) ˆ ( − \\ compute Z( 2m)Z = 0 . 5 ∗ ( − ∗ (A ∗ B − C+D ∗ (P1+P2 ) ) ; \\ compute Zn up t o nth − i f ( n==1,Zn=0 , for ( j =1 ,n − ∗ m) ) ) ; \\ compute and p r i n t tntn [ n ] = (Z − Zn)ˆ( − ∗ m) ) ;p r i n t (m, ” : ” , tn [ n ] ) ;m = m+step m ; \\ d e c r e a s e m by step mn = n+1; \\ i n c r e m e n t z e r o c o u n t e r) } Listing 4: PARI script for generating non-trivial zeros from primes.26very iteration. By leveraging these parameters, the output can converge todifferent values, and in some cases will not converge. We optimized them togive 4 zeros accurately, and beyond that it doesn’t converge and then m has tobe increased to a larger value. The results of running this script are summarizedin Table 6. As before, we obtain t accurate to 87 decimal places, but t nowis accurate to 28 decimal places, and the next zero to 12 and 1 decimal placesrespectively. At this point the iteration has ran its course. We would like toincrease m , but presently is outside the range of the test computer.Table 6: The t n +1 by PARI scrip in Listing 4 m n t n +1 First 30 digits of computed results Significant Digits250 0 t t t t4