Analytical recurrence formulas for non-trivial zeros of the Riemann zeta function
aa r X i v : . [ m a t h . N T ] D ec Analytical recurrence formulas for non-trivialzeros of the Riemann zeta function
Artur Kawalec
Abstract
In this article, we develop four types of analytical recurrence formulasfor non-trivial zeros of the Riemann zeta function on critical line assuming(RH). Thus, all non-trivial zeros up to the n th order must be known inorder to generate the n th+1 non-trivial zero. All the presented formulasare based on a certain closed-form representations of the secondary zetafunction family which are already available in the literature. We alsopresent a formula to generate the non-trivial zeros directly from primes,thus all primes can be converted into an individual non-trivial zero. Andsimilarly, we can convert all non-trivial zeros into an individual prime.We also extend the presented results to other Dirichlet-L functions, andin particular, we develop an analytical recurrence formula for non-trivialzeros of the Dirichlet beta function. Throughout this article, we alsonumerically compute these formulas to high precision for various test casesand review the computed results. The Riemann zeta function is classically defined by an infinite series ζ ( s ) = ∞ X n =1 n s (1)which is absolutely convergent for ℜ ( s ) >
1, where s is a complex variable s = σ + it . By analytically continuing (1) to the whole complex plane, thefunction has an infinity of complex non-trivial zeros constrained to lie in acritical strip region 0 < σ <
1. The n th zero is denoted as ρ n = σ n + it n and isa solution of ζ ( ρ n ) = 0 (2)for n ≥
1. The first few zeros on the critical line at σ n = have imaginarycomponents t = 14 . ... , t = 21 . ... , t = 25 . ... , andso on, which were originally computed numerically using an equation solver,1ut if the Riemann Hypothesis (RH) is true, then can be represented by ananalytical recurrence formula as t n +1 = lim m →∞ " ( − m (cid:18) m − m − | ζ | ) (2 m ) (cid:0) (cid:1) − m ζ (2 m,
54 ) (cid:19) − n X k =1 t mk − m (3)that we developed in [7], thus all non-trivial zeros up to the n th order mustbe known in order to generate the n th+1 non-trivial zero. We consider the2 m limiting value as to ensure that it is even. The key component of thisrepresentation is a termlog( | ζ | ) (2 m ) (cid:0) (cid:1) = d m ds m log( | ζ ( s ) | ) (cid:12)(cid:12)(cid:12) s = (4)which is the 2 m -th derivative of log[ ζ ( s )] evaluated at s = , which can also bewritten as a Cauchy integral aslog( | ζ | ) (2 m ) (cid:0) (cid:1) = (2 m )!2 πi I Ω log( | ζ ( z ) | )( z − ) m +1 dz (5)where Ω is a small contour encircling a point s = . Also, ζ ( s, a ) is the Hurwitzzeta function ζ ( s, a ) = ∞ X n =0 n + a ) s , (6)which is a shifted version of (1) by an arbitrary parameter a > m th logarithmic derivatives of a certain function.In no particular order, we define several variations of the secondary zetafunctions. The first kind is a generalized zeta series over imaginary parts ofnon-trivial zeros defined by Z ( s ) = ∞ X n =1 t sn . (7)The second kind is a generalized zeta series over complex non-trivial zeros takenin conjugate-pairs defined as Z ( s ) = ∞ X n =1 ρ sn = ∞ X n =1 " + it n ) s + 1( − it n ) s . (8)2he value for Z (1) = 1 + 12 γ −
12 log(4 π ) (9)is commonly known throughout the literature [4, p.67]. The third kind is ageneralized zeta series over the complex magnitude, or modulus, squared ofnon-trivial zeros defined as Z ( s ) = ∞ X n =1 | ρ n | s = ∞ X n =1 + t n ) s . (10)The forth kind is a generalized Jacobi theta series for the sum of exponentialsover the imaginary parts of non-trivial zeros defined by Z ( s ) = ∞ X n =1 e − t n s . (11)In this article, we consider a case when s → ∞ and use the closed-form rep-resentations of these family of secondary zeta functions that is already availablein the literature, and extract non-trivial zeros under the right excitation of theseseries. In the previous paper [7], we developed a formula for non-trivial zeros(3) based on the secondary zeta function of the first type (7), which we willreview in Section 2. In Section 3, we develop another independent formula fornon-trivial zeros given by Matsuoka [9] based on combining the secondary zetafunctions for (8) and (10), and in Section 4, we further expand on Section 3 anddevelop another representation for non-trivial zeros which can be obtained di-rectly from primes, thus we can convert all primes into an individual non-trivialzero. And conversely, we also give a set of formulas to convert all non-trivialzeros into an individual prime. In Section 5 we outline another formula basedon the Jacobi type of secondary zeta function (11). And in Section 6, we extendthese formulas to non-trivial zeros of the Dirichlet-L function, and in particular,we develop a similar formula to (3) for the non-trivial zeros of the Dirichlet betafunction defined as β ( s ) = ∞ X n =0 ( − n (2 n + 1) s , (12)which is convergent for ℜ ( s ) > β ( s ), the convergence to its zeros issignificantly slower than for ζ ( s ), and for the Jacobi generalized series (11), thecomputational requirements are presently outside the range of the test computerto compute directly. 3 A formula for non-trivial zeros of the first kind
We consider the secondary zeta function of the first type as Z ( s ) = ∞ X n =1 t sn = 1 t s + 1 t s + 1 t s + . . . , (13)which is convergent for ℜ ( s ) >
1. The special values for the first few positiveintegers is 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= 0 . . . .,Z (5) = 0 . . . .. (14)The values for even integer argument are given by a closed-form formula for Z (2 m ) as 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) (15)assuming (RH) and which is valid for a positive integer m ≥
1, but the 2 m ensures that the limit variable is even. Although this formula looks relativelysimple, it is actually a result of a complicated and detailed analysis by works ofVoros [11][12] by analytically continuing (13) to the whole complex plane usingMellin transforms and tools from spectral theory. There is no known closed-form formula such as this that is valid for odd integer argument, the odd valuesgiven were computed using a special algorithm that was developed by Arias DeReyna [2] (further elaborated in Appendix A) and is available as a stand-alonefunction in a Python library. It would otherwise take billions of zeros to reachsuch accuracy for the odd values.Furthermore, we also have a useful identity12 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) , (16)4lso found in [11, p.681], in which we can express the zeta and beta terms interms of a Hurwitz zeta function resulting in a compact form Z (2 m ) = ( − m (cid:20) m − m − | ζ | ) (2 m ) (cid:0) (cid:1) − m ζ (2 m,
54 ) (cid:21) . (17)To find the non-trivial zeros, we consider solving for t in (13) to obtain1 t s = Z ( s ) − t s − t s − . . . (18)and then we get t = (cid:18) Z ( s ) − t s − t s − . . . (cid:19) − /s . (19)If we then consider the limit t = lim s →∞ (cid:18) Z ( s ) − t s − t s − . . . (cid:19) − /s (20)then, 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 . (21)Hence, by substituting (17) into (21), the formula for t becomes t = lim m →∞ (cid:20) ( − m (cid:18) m − m − | ζ | ) (2 m ) (cid:0) (cid:1) − m ζ (2 m,
54 ) (cid:19)(cid:21) − m . (22)Next we numerically verify this formula in PARI, and the script is shown inListing 1. We broke up the representation (22) 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 1 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 . In Appendix B, we also give a secondscript in the Mathematica software package to compute (22) using the Cauchyintegral (5).Next we numerically compute (22) for m = 250 which yields t = 14 . . . . (23)which is accurate to 87 digits. 5able 1: The computation of t by equation (3) 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 34 { \\ 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 = 2 ˆ (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 1: PARI script for computing equation (3).In order to find the second non-trivial zero, we comeback to (13), and solvingfor t yields t = lim s →∞ (cid:18) Z ( s ) − t s − t s − . . . (cid:19) − /s (24)6nd since the higher order zeros decay as t − s faster than Z ( s ) − t − s , we thenhave t = lim s →∞ (cid:18) Z ( s ) − t s (cid:19) − /s (25)as O ( t − s ) vanishes, and the zero becomes t = lim m →∞ (cid:20) ( − m (cid:18) m − m − | ζ | ) (2 m ) (cid:0) (cid:1) − m ζ (2 m,
54 ) (cid:19) − t m (cid:21) − m . (26)A numerical computation for m = 250 yields t = 21 . . . . (27)which 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 limit variable as it will cause self-cancelation in(26), and also 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 →∞ (cid:20) ( − m (cid:18) m − m − | ζ | ) (2 m ) (cid:0) (cid:1) − m ζ (2 m,
54 ) (cid:19) − t m − t m (cid:21) − m (28)since Z ( s ) − t − s − t − s ≫ t − s . A numerical computation for m = 250 yields t = 25 . . . . (29)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. As aresult, if we define a partial secondary zeta function up to the n th order Z ,n ( s ) = n X k =1 t sk , (30)then the n th+1 non-trivial zero is t n +1 = lim m →∞ [ Z ( m ) − Z ,n ( m )] − /m (31)because t − sn ≫ t − sn +1 , and the main formula: t n +1 = lim m →∞ " ( − m (cid:18) m − m − | ζ | ) (2 m ) (cid:0) (cid:1) − m ζ (2 m,
54 ) (cid:19) − n X k =1 t mk − m . (32)7ne can actually use any number of representations for Z ( s ), and the chal-lenge will be find more efficient algorithms to compute them. And finally, wereport a numerical result for Z (500) as: Z = 7 . × − . . . . (33)From this number, we extracted the first 10 non-trivial zeros, which are summa-rized in Table 2 for m = 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 with same limit variable becauseit will cause self-cancelation, and the accuracy for t n must be much higher than t n +1 to ensure convergence. Initially we started with an accuracy of 87 digitsafter decimal place for t , and then it dropped to 7 to 12 digits by the time itgets to t zero. There is also a sudden drop in accuracy when the gaps get toosmall. And as the gap gets too small, it has to be overcompensated by increas-ing m higher. Hence, these formulas are not very practical for computing highzeros as large numerical precision is required, especially when we get to the firstLehmer pair at t = 7005 . ∼ . t n +1 − t n ∼ π log( n ) , makingthe use of this formula progressively harder and harder to compute.Table 2: The t n +1 computed by equation (3). n t n +1 m = 250 Significant Digits0 t t t t t t t t t t A formula for non-trivial zeros of the secondkind
The secondary zeta function of the second kind as defined by (8) is Z ( s ) = ∞ X n =1 ρ sn = ∞ X n =1 " + it n ) s + 1( − it n ) s , (34)where the sum is taken over reciprocal zeros taken in conjugate-pairs raised topower s . The first few special values of this series are: Z (1) = 1 + 12 γ −
12 log(4 π )= 0 . . . .,Z (2) = 1 + γ + 2 γ − π = − . . . .,Z (3) = 1 + γ + 3 γγ + 32 γ − ζ (3)= − . . . .,Z (4) = 1 + γ + 4 γ γ + 2 γ + 2 γγ + 23 γ − π = 0 . . . .,Z (5) = 1 + γ + 5 γ γ + 52 γ γ + 52 γ γ + 5 γγ + 56 γγ + 524 γ − ζ (5)= 0 . . . .. (35)The value for Z (1) is commonly known throughout the literature and valuesfor Z ( m ) for m > Z ( m ) = 1 − ( − m − m ζ ( m ) − log( | ζ | ) ( m ) (0)( m − m > m .Another representation of (36) is given by Z ( m ) = 1 − (1 − − m ) ζ ( m ) + g cm ( m − g cm is a Stieltjes Cumulants defined by Voros in [12, p.25], which are theexpansion coefficients of a certain serieslog[( s − ζ ( s )] = − ∞ X n =1 ( − n n ! g cn ( s − s , (38)as to conveniently extract the n th derivative at s = 0. This series is not com-monly known, but the very well-known Laurent expansion coefficients for theseries ζ ( s ) = 1 s − ∞ X n =0 ( − n n ! γ n ( s − n (39)are the Stieltjes constants, and the value for γ = γ = 0 . . . . is theEuler-Mascheroni constant. We also define yet another series − ζ ′ ( s ) ζ ( s ) = 1 s − ∞ X n =0 ( − n n ! η n ( s − n , (40)where η n are its Laurent expansion coefficients. The relation between η n and γ n is given by a recurrence relation η n = ( − n +1 " n + 1 n ! γ n + n − X k =0 ( − k − ( n − k − η k γ n − k − (41)found in Coffey [3, p.532] and then g cn is g cn = ( − n ( n − η n − (42)found in [12, p.25]. Essentially, all of these coefficients are variants of oneanother. Now, by substituting g cn to equation (37) can now generate the valuesfor Z ( m ) in terms of the Stieltjes constants in (35) and as shown on Wolfram’swebsite [13].Now, when trying to extract the non-trivial zeros using Z ( s ) we encounterdifficulty when combining the conjugate-pairs of zeros as shown w n ( s ) = 1( + it n ) s + 1( − it n ) s , (43)where w n ( s ) is an n th conjugate-pair term. From this form, it is not readilypossible to separate the non-trivial zero terms in the limit as s → ∞ similarly asin the non-trivial zero formula (3). The solution to this was given by Matsuoka[9], and what we can do is a slightly different interpretation, first we find w n ( s ) = 2( + t n ) s + 1( + it n ) s + 1( − it n ) s , (44)then we get 1 | ρ n | s = 1( + t n ) s = 12 [ w n ( s ) − w n (2 s )] . (45)10his motivates to define a new secondary zeta function for the reciprocal powersof complex magnitude, or modulus, squared of ρ as Z ( s ) = ∞ X n =1 | ρ n | s = ∞ X n =1 + t n ) s = ∞ X n =1
12 [ w n ( s ) − w n (2 s )]= ∞ X n =1 w n ( s ) − Z (2 s ) . (46)We then need to find another formula for w n ( s ) which is an n th conjugate-pairterm squared as defined above. If we expand (34) as Z ( s ) = ∞ X n =1 w n ( s ) = w ( s ) + w ( s ) + w ( s ) + . . . (47)so that Z ( s ) = ∞ X n =1 w n ( s ) = w ( s ) + O [2 w ( s ) w ( s )] , (48)because | w ( s ) | ≫ | w ( s ) || w ( s ) | as s → ∞ . Now, substituting (48) to (46)yields an asymptotic formula for Z ( s ) ∼
12 [ Z ( s ) − Z (2 s )] (49)as s → ∞ , which is actually what we need to extract non-trivial zeros. Whatwe don’t have is a formula for Z ( s ) for an arbitrary s , but that is not neededfor the next step. Hence, if we begin with the secondary zeta function Z ( s ) = 1( + t ) s + 1( + t ) s + 1( + t ) s + . . . (50)and then solving for t we obtain1( + t ) s = Z ( s ) − + t ) s − + t ) s − . . . (51)and then we get14 + t = (cid:20) Z ( s ) − + t ) s − + t ) s − . . . (cid:21) − /s (52)and this leads to t = "(cid:18) Z ( s ) − + t ) s − + t ) s − . . . (cid:19) − /s − / . (53)11f we then consider the limit as s → ∞ , then the higher order terms decay as O [(1 / t ) − s ], and hence, substituting equation (49) for Z ( s ) yields t = lim s →∞ "(cid:18) Z ( s ) − Z (2 s ) (cid:19) − /s − / (54)which was given by Matsuoka in [9]. One can substitute any representation of Z ( s ) such as by equations (36) or (37), which involves the Stieltjes constants.Next we numerically verify this formula in PARI, and the script is shown inListing 2. We use equation (36) for Z ( s ) and broke up the representation (54)into several parts A to C. And as before, sufficient memory must be allocatedand precision set to high before running the script. The results are summarizedin Table 3 for various limit values of m from low to high, and we can observe theconvergence to the real value as m increases. Already at m = 10 we get severaldigits of t , and at m = 100 we get over 16 digits. We observe that for oddvalue of m the convergence is slightly better than for even m . We performedhigher precision computations, and the result is clearly converging to t .Next we numerically verify it for m = 250 which yields t = 14 . . . . (55)which is accurate to 43 digits. Henceforth, in order to find the second non-trivialzero, we comeback to (50), and solving for t yields t = "(cid:18) Z ( s ) − + t ) s − + t ) s . . . (cid:19) − /s − / (56)and since the higher order zero terms decay as + t ) s faster than Z ( s ) − + t ) s ,we then have t = lim s →∞ "(cid:18) Z ( s ) − Z ( s ) − + t ) s (cid:19) − /s − / . (57)And continuing on, the next zero is computed as t = lim s →∞ "(cid:18) Z ( s ) − Z (2 s ) − + t ) s − + t ) s (cid:19) − /s − / (58)since the higher order zero terms decay as + t ) s . As a result, if we define apartial secondary zeta function up to the n th order Z ,n ( s ) = n X k =1 | ρ k | s = n X k =1 + t k ) s (59)12able 3: The computation of t by equation (54) for different m .m t (First 30 Digits) Significant Digits2 5.561891787634141032446012810136 03 13.757670503723662711511861003244 04 12.161258748655529488677538477512 05 14.075935317783371421926582853327 06 13.579175424560852302300158195372 07 14.116625853057249358432588137893 18 13.961182494234115467191058505224 09 14.126913415083941105873032355837 110 14.077114859427980275510456957007 015 14.133795710050725394699252528681 220 14.134370485636531946259958638820 325 14.134700629574414322701677282886 450 14.134725141835685792188021492482 9100 14.134725141734693789329888107217 16 { \\ s e t l i m i t v a r i a b l em1 = 2 5 0 ; \\ compute p a r a m e t e r s A1 t o C1 f o r Z1A1 = derivnum ( x=0 , l o g ( abs ( z e t a ( x ) ) ) , m1 ) ;B1 = 1/ f a c t o r i a l (m1 − − ( − ∗ − m1) ∗ z e t a (m1 ) ;Z1 = C1 − A1 ∗ B1 ; \\ compute 2m l i m i t v a r i a b l em2 = 2 ∗ m1 ; \\ compute p a r a m e t e r s A2 t o C2 f o r Z2A2 = derivnum ( x=0 , l o g ( abs ( z e t a ( x ) ) ) , m2 ) ;B2 = 1/ f a c t o r i a l (m2 − − ( − ∗ − m2) ∗ z e t a (m2 ) ;Z2 = C2 − A2 ∗ B2 ;t 1 = ( ( ( Z1ˆ2 − Z2 )/2)ˆ( − − } Listing 2: PARI script for computing equation (54).13hen the n th+1 non-trivial zero is t n +1 = lim s →∞ [ Z ( s ) − Z ,n ( s )] − /s (60)and the main recurrence formula: t n +1 = lim s →∞ Z ( s ) − Z (2 s ) − n X k =1 + t k ) s ! − /s − / . (61)Next, when we attempt to numerically verify (61) for higher zeros starting witha limit variable m = 250, then we get t accurate to 43 decimal places as before,however, such precision is not enough to be able to compute t , so we have toincrease the limit variable m to achieve higher precision which presently is atthe limit of our test computer. We did, however, verify (61) successfully bypre-computing Z ( s ) using 100 non-trivial zeros known to high precision (2000decimal places), and then we computed the next zeros by (61), but presently,limitations of the test computer prevent computing Z ( s ) using (49) to highenough precision. In this section, we develop a variation a formula for non-trivial zeros based onprimes. We define a (Hurwitz) shifted version of Z ( s ) by a parameter a as Z ( s | a ) = ∞ X n =1 ρ n + a − ) s . (62)The usual Z ( s ) as defined by equation (8) is a special case for a = . But when a > , there is another closed-form representation Z ( s | a ) = ( a −
12 ) − s − − s ζ ( s,
54 + 12 a ) − s ) ∞ X k =2 Λ( k ) k + a (log k ) s − (63)found in Voros [12, p.56] which involves the von Mangoldt’s function:Λ( n ) = ( log p, if n = p k for some prime and integer k ≥ , . (64)Now, if we apply the same arguments as in Section 3 to extract the non-trivialzeros, we obtain the first zero t = lim s →∞ "(cid:18) Z ( s | a ) − Z (2 s | a ) (cid:19) − /s − a / (65)14nd the full recurrence formula: t n +1 = lim s →∞ Z ( s | a ) − Z (2 s | a ) − n X k =1 a + t k ) s ! − /s − a / . (66)Hence through these formulas, the primes are directly converted into non-trivialzeros by an infinite series involving the Λ( k ) and also the Hurwitz zeta function.This formula is valid for an arbitrary parameter a > , but we find numericallythat the convergence is very slow due to the nature of the von Mangoldt’s func-tion series, which requires billions of primes to reach some reasonable accuracy.When we test this formula, we find that convergence is improved when a isincreased but not too much in relation to the limit variable s . The script inPARI is shown in Listing 3, and we run equation (65) with parameters k = 10 (up to a billionth value for Λ( k )) and s = 50 and a = 15. The result is: t = 14 . . . . (67)which is accurate to 4 digits.As shown in [7], we also outline the duality between primes and non-trivialzeros. The formula (66) converts all primes into an individual non-trivial zero.To complete the duality, it is also possible to convert all non-trivial zeros intoan individual prime by means of the Golomb’s formula for primes and theHadamard product for ζ ( s ). Let p = 2, p = 3, p = 5 and so on, definea sequence of primes, and Q n ( s ) define a partial Euler prime product up the n th order Q n ( s ) = n Y k =1 (cid:18) − p sk (cid:19) − (68)for n > Q ( s ) = 1, then the Golomb’s recurrence formula for the p n +1 prime is p n +1 = lim s →∞ [ ζ ( s ) − Q n ( s )] − /s (69)as shown in [5] and [6]. And since ζ ( s ) can be written in terms of the Hadamardproduct in terms of non-trivial zeros ζ ( s ) = π s/ s − s ) ∞ Y k =1 (cid:18) − sρ k (cid:19) , (70)we can substitute (70) to (69) and obtain p n +1 = lim s →∞ " π s/ s − s ) ∞ Y k =1 (cid:18) − sρ k (cid:19) − Q n ( s ) − /s (71)15nd the full recurrence formula: p n +1 = lim s →∞ " π s/ s − s ) ∞ Y k =1 (cid:18) − sρ k (cid:19) − n Y k =1 (cid:18) − p sk (cid:19) − − /s . (72)Hence, this is a way from converting non-trivial zeros to the primes and withoutassuming (RH), as the Hadamard product is over all zeros.Mangoldt ( n)= { i s p o w e r ( n , ,& n ) ;i f ( i s p r i m e ( n ) , l o g ( n ) , 0 ) }{ \\ s e t v a r i a b l e ss1 = 5 0 ; \\ l i m i t v a r i a b l ea =15; \\ a r b i t r a r y pa r a meterk = 1 0 0 0 0 0 0 0 0 0 ; \\ von Mangoldt sum l i m i t \\ compute p a r a m e t e r s A t o C f o r Z1y1 = sum ( n=2 ,k , Mangoldt ( n ) / nˆ(1/2+ a ) ∗ l o g ( n ) ˆ ( s1 − − − s1 ) − − s1 ) ∗ z e t a h u r w i t z ( s1 ,5/4+1/2 ∗ a) − ∗ y1 ; \\ compute do uble l i m i t v a r i a b l es2 = 2 ∗ s1 ; \\ compute p a r a m e t e r s A t o C f o r Z2y2 = sum ( n=2 ,k , Mangoldt ( n ) / nˆ(1/2+ a ) ∗ l o g ( n ) ˆ ( s2 − − − s2 ) − − s2 ) ∗ z e t a h u r w i t z ( s2 ,5/4+1/2 ∗ a ) − ∗ y2 ; \\ compute t 1t 1 = ( ( ( Z1ˆ2 − Z2 )/2)ˆ( −
1/ s1 ) − a ˆ 2 ) ˆ ( 1 / 2 ) ;p r i n t ( t 1 ) ; } Listing 3: PARI script for computing equation (65).16
Non-trivial zeros from infinite series over ex-ponentials
In this section, we explore yet another formula for non-trivial zeros. The Jacobigeneralized series over the exponentials of t n is defined by Z ( s ) = ∞ X n =1 e − t n s = e − t s + e − t s + e − t s . . . , (73)which has a closed-form representation given by Z ( s ) = A ( s ) − B ( s ) , (74)where A ( s ) = − √ πs ∞ X k =2 Λ( k ) √ k e − s log k + e s (75)and B ( s ) = γ + log(16 π s )8 √ πs − √ πs Z ∞ e − u s u − e u e u − ! du (76)which is given in [2, p.3]. It is seen that it also involves the von Mangoldt’sfunction and hence the primes. The terms of this series decay extremely fastdue to the exponential nature. The first term is Z ( s ) ∼ O ( e − t s ) (77)so that is suffices to solve for t and we get t = lim s →∞ r − s log Z ( s ) (78)and the recurrence formula is t n +1 = lim s →∞ vuut − s log Z ( s ) − n X k =1 e − t k s ! . (79)Next we numerically compute equation (73) for s = 2 by summing the firsttwo zeros to obtain Z ( s ) = 2 . × − . . . . (80)The result gets extremely small as s increases, and hence the first term involving t dominates the series. Now, if we compute using equation (74) (the script isnot given) with Λ( n ) summed to k = 10 , then we get A = 0 . . . . (81)17nd B = 0 . . . . (82)and Z ( s ) = A − B = 2 . × − . . . . (83)We observe that the difference between A and B must result in a cancelationoccurring very far out in the decimal places in order to converge to (80), whichis presently outside of reach of the test computer. The Λ( n ) series in A is veryslow to converge, while the integral term in B is much faster. Hence this formulais not practical, and is presently outside the range of what we can compute, butin principle should yield the non-trivial zeros. The Dirichlet beta functions as defined as β ( s ) = ∞ X n =0 ( − n (2 n + 1) s , (84)which is convergent for ℜ ( s ) >
0. It is useful to define β ( s ) = 14 s [ ζ ( s,
14 ) − ζ ( s,
34 )] (85)in terms of the Hurwitz zeta function since it is available in most mathematicalsoftware packages where it can be efficiently computed, except at s = 1 whereit has a pole, but it could be handled in a limiting sense s →
1. The value for β (1) = π .Let r n be imaginary components of non-trivial zeros of β ( s ) on the criticalline. The first few non-trivial zeros on the critical line have imaginary compo-nents r = 6 . ... , r = 10 . ... , r = 12 . ... which wereoriginally computed numerically, but now can also be computed analyticallyby essentially the same arguments as described in Section 2. If we define thesecondary beta function B ( s ) = ∞ X k =1 r sk (86)so that B ( s ) is a sum of reciprocal powers of imaginary components of non-trivial zeros. Then, we consider a partial secondary beta function up to the n thorder B n ( s ) = n X k =1 r sk , (87)and because r − s ≫ r − s ≫ r − s ≫ r − sn . . . as s → ∞ , then the non-trivial zerosare given by a recursive relationship r n +1 = lim s →∞ [ B ( s ) − B n ( s )] − /s . (88)18t now suffices to find a suitable formula for B ( s ) which is also given by Vorosin [12, p.110] as a general formula for Dirichlet-L functions. If we take L χ to be β ( s ) then we have B (2 m ) = −
12 [2 m − ζ (2 m ) + (1 − a )2 m β (2 m )] − m − | β | ) (2 m ) ( 12 )(89)assuming (GRH) for β ( s ). The parity parameter a is related to the Dirichletcharacter, which we take it to be a = 1. We further obtain B (2 m ) = ( − m +1 (cid:20) m − [(1 − − m ) ζ (2 m ) − β (2 m )] + 1(2 m − | β | ) (2 m ) ( 12 ) (cid:21) . (90)Since the zeta term in (90) above is related to Dirichlet lambda function(1 − − m ) ζ (2 m ) = λ (2 m ) = ∞ X n =0 n + 1) m (91)we can simplify this further, and obtain(1 − − m ) ζ (2 m ) − β (2 m ) = 2 ∞ X n =1 n − m = 12 m − ζ (2 m,
34 ) (92)which leads to a more compact form B (2 m ) = ( − m +1 (cid:20) m − | β | ) (2 m ) ( 12 ) + 12 m ζ (2 m,
34 ) (cid:21) . (93)This results in a direct formula for r as r = lim m →∞ (cid:20) ( − m +1 (cid:18) m − | β | ) (2 m ) ( 12 ) + 12 m ζ (2 m,
34 ) (cid:19)(cid:21) − m (94)and a full recurrence formula: r n +1 = lim m →∞ " ( − m +1 (cid:18) m − | β | ) (2 m ) ( 12 ) + 12 m ζ (2 m,
34 ) (cid:19) − n X k =1 r mk − m . (95)A script to compute r is presented in Listing 4, and the computed values forvarious limit values of m from low to high is shown in Table 4, where we observea very slow convergence to r . At m = 250, we get 1 digit of accuracy, whichpresently is the limit of what we can compute.19able 4: The computation of r by equation (94) for different m .m r (First 30 Digits) Significant Digits2 4.816777631972992790199023658953 03 5.315392096740758247236038034750 04 5.509658064544543604515877021145 05 5.614693890956563572345457016812 06 5.682164176401417068175722897838 07 5.729858492194765348617713490968 08 5.765605628007070123200909419564 09 5.793477676180773529979653964463 010 5.815846178131514981311063955026 015 5.883430218269397015707767675913 020 5.917512606238506379951650937763 025 5.938056721895896708189172825057 050 5.979359172665024315170697408792 0100 6.000118003960505930982234572329 1200 6.010524430031237972442214363982 1250 6.012607880075946303077925471894 1 \\ D e f i n e D i r i c h l e t beta f u n c t i o nbeta ( x)= { − x ∗ ( z e t a h u r w i t z ( x ,1/4) − z e t a h u r w i t z ( x , 3 / 4 ) ) ; }{ \\ 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 ( beta ( x ) ) , 2 ∗ m) ;B = 1/ f a c t o r i a l ( 2 ∗ m − − ∗ m) ∗ z e t a h u r w i t z ( 2 ∗ m, 3 / 4 ) ; \\ compute B( 2m)Z = ( − ∗ (A ∗ B+C ) ; \\ compute r 1r 1 = Zˆ( − ∗ m) ) ;p r i n t ( r 1 ) ; } Listing 4: PARI script for computing equation (94).20 eferences [1] M. Abramowitz and I. A. Stegun.
Handbook of Mathematical Functionswith Formulas, Graphs, and Mathematical Tables . Dover Publications, ninthprinting, New York, (1964).[2] J. Arias De Reyna.
Computation of the secondary zeta func-tion .math.NT/arXiv:2006.04869, (Jun. 2020).[3] M. Coffey.
Relations and positivity results for the derivatives of the Riemann ξ function . Elsevier, Journal of Computational and Applied Mathematics.166, 525-534, (2004).[4] H.M. Edwards. Riemann’s Zeta Function . Dover Publications, Mineola, NewYork (1974).[5] S. Golomb.
Formulas for the next prime . Pacific Journal of Mathematics, ( The n th+1 prime limit formulas .math.GM/arXiv:1608.01671v2,(Aug. 2016).[7] A. Kawalec. The recurrence formulas for primes and non-trivial zeros of theRiemann zeta function . math.GM/arXiv:2009.02640, (Sep. 2020).[8] D.H. Lehmer.
The Sum of Like Powers of the Zeros of the Riemann ZetaFunction . Mathematics of Computation. (181), 265-273, (1988).[9] Y. Matsuoka. A sequence associated with the zeros of the Riemann zeta func-tion . Tsukuba J. Math. (2), 249-254, (1986).[10] The PARI Group, PARI/GP version , Univ. Bordeaux, (2019).[11] A. Voros. Zeta functions for the Riemann zeros . Ann. Institute Fourier, ,665–699,(2003).[12] A. Voros. Zeta Functions over Zeros of Zeta Functions . Springer; 2010thedition (2010)[13] Weisstein, Eric W.
Riemann Zeta Function Ze-ros . From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannZetaFunctionZeros.htmlEmail: [email protected] Appendix A
There is an available algorithm developed by Arias De Reyna as described in [2]to compute Z ( s ) and which is fully implemented in a Python library mpmath as a secondary zeta function and is analytically extended to the whole complexplane. Roughly, the algorithm takes a finite number of non-trivial zeros anda finite number of primes for the von Mangoldt’s function term and estimatesthe remainder yielding an accurate computation of Z ( s ) to a high number ofdecimal places. We tested this function and computed the odd values for Z ( s )accurately as shown in (14), which otherwise would take billions of zeros toreach. The Listing 5 shows the script, and in Table 5, we compare the resultsfor even values for Z ( s ) with reference values computed by the closed-formformula (3) as Z ,ref in PARI with precision set to 2000 decimal places andgive the values to the first 30 decimal places. Also, the precision in Python wasset to 100 decimal places. The remainder was compared | Z ,ref − Z ,python | .We observe that the remainder for even values as given by the Python scriptmatches values computed by the closed-form formula, and in fact exceeds in allthe cases by at least an order of magnitude. Therefore we conclude that thegiven odd values should also be accurate to within that remainder, in fact, Z ( s )can be computed for any complex s , which is further explored in [2]. from mpmath import ∗ mp. p r e t t y = True ; mp. dps = 100z = s e c o n d z e t a ( 3 , e r r o r=True )Listing 5: Python script for computing Z ( s ) by the algorithm in [2].Table 5: The first 30 digits Z ( m ) computed using the Python script withdefault parameter a = 0 . m Z ( m ) Remainder | Z ,ref − Z ,python | − − − -4 0.000037172599285269686164866262 10 − − − -6 0.000000144173931400973279695381 10 − − − -8 0.000000000663031680252990869873 10 − − − -10 0.00000000000321366415061660121 10 − −
11 0.00000000000022556506251559664 10 − -22 Appendix B
In this section, we compute t in Mathematica using Z ( s ) by equation (3), butinstead of computing the m th derivative, we compute the Cauchy integral takenalong a closed contour Ω which is a square loop with 0 . s = . The script is shown in Listing 6. With m = 50 we obtain t = 14 . . . . (96)which is accurate to 16 decimal places. ( ∗ S e t l i m i t v a r i a b l e ∗ ) m = 5 0 ; ( ∗ Compute Cauchy i n t e g r a l around omega ∗ ) K = Factori al [ 2 m] / ( 2 \ [ Pi ] I ) ; ( ∗ D e f i n e i n t e g r a n d ∗ ) F [ z ] :=
Log [ Zeta [ z ] ] / ( z − ( ∗ I n t e g r a t e around a c l o s e d l o o p ∗ ) I 1 =
NIntegrate [ F [ z ] , { z , 0 . 7 5 − I , 0 . 7 5 + 0 . 2 5 I } , WorkingPrecision − > NIntegrate [ F [ z ] , { z , 0 . 7 5 + 0 . 2 5 I , 0 . 2 5 + 0 . 2 5 I } , WorkingPrecision − > NIntegrate [ F [ z ] , { z , 0 . 2 5 + 0 . 2 5 I , 0 . 2 5 − I } , WorkingPrecision − > NIntegrate [ F [ z ] , { z , 0 . 2 5 − I , 0 . 7 5 − I } , WorkingPrecision − > K ( I 1 + I 2 + I 3 + I 4 ) ; ( ∗ Compute Z ∗ ) A = I x /
Factori al [ 2 m − − − − A − B ) ; ( ∗ Compute t 1 ∗ ) t 1 = Zˆ( − t1