A Concise Proof of the Riemann Hypothesis Based on Hadamard Product Fayez Alhargan

To cite this version:

Fayez Alhargan. A Concise Proof of the Riemann Hypothesis Based on Hadamard Product. 2021. hal-01819208v5

HAL Id: hal-01819208 https://hal.archives-ouvertes.fr/hal-01819208v5 Preprint submitted on 7 Jul 2021

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A CONCISE PROOF OF THE RIEMANN HYPOTHESIS BASED ON HADAMARD PRODUCT∗

FAYEZ A. ALHARGAN†

Abstract. A concise proof of the Riemann Hypothesis is presented by clarifying the Hadamard product expansion over the zeta zeros, demonstrating conclusively that the Riemann Hypothesis is true. Then, an accurate zero-counting function exhibiting the expected step function behaviour is developed. Also, based on the Heaviside function a unifying analysis of the prime-counting function is presented and its relation to the zeta function is developed via the Laplace transform and the residue theorem. Furthermore, a new s-domain deﬁnitions of the prime-counting function and the Chebyshev function are developed, revealing a profound underlining relationship to the zeta function. The paper encompass a new paradigm shift in the prime analysis with a fresh perspective from the s-domain.

Key words. the Riemann Hypothesis, the functional equation, the Riemann zeta function, Hadamard Product

AMS subject classiﬁcations. 11M26

1. Introduction. In his landmark paper in 1859, Bernhard Riemann [1] hypoth- esized that the non-trivial zeros of the Riemann zeta function ζ(s) all have a real part 1 equal to 2 . Major progress towards proving the Riemann hypothesis was made by Jacques Hadamard in 1893 [2], when he showed that the Riemann zeta function ζ(s) can be expressed as an infinite product expansion over the non-trivial zeros of the zeta function. In 1896 [3], he also proved that there are no zeros on the line <(s) = 1. The Riemann Hypothesis is the eighth problem in David Hilbert’s list of 23 unsolved problems published in 1900 [4]. There has been tremendous work on the subject since then, which has been illustrated by Titchmarsh (1930) [5], Edwards (1975) [6], Ivic (1985) [7], and Karatsuba (1992) [8]. It is still regarded as one of the most diffi- cult unsolved problems and has been named the second most important problem in the list of the Clay Mathematics Institute Millennium Prize Problems (2000), as its proof would shed light on many of the mysteries surrounding the distribution of prime numbers [9, 10]. The Riemann zeta function is a function of the complex variable s, defined in the half-plane <(s) > 1 by the absolutely convergent series

∞ X 1 (1.1) ζ(s) := ns n=1 and in the whole complex plane by analytic continuation [9]. The Riemann hypothesis is concerned with the locations of the non-trivial zeros 1 of ζ(s), and states that: the non-trivial zeros of ζ(s) have a real part equal to 2 [9]. In this paper, the truth of the Riemann Hypothesis is demonstrated by employing the Hadamard product of the zeta function and clarifying the principle zeros for the product expansion. The process is outlined in a less abstract form, to be accessible for a wider audience. Furthermore, new concise and accurate results have been obtained for the prime- counting function and the zero-counting function. These were achieved by utilizing new techniques; that included the Heaviside function, Dirac delta function, the

∗Revised Version Submitted to the editors 5 March 2021. †PSDSARC, Riyadh, Saudi Arabia ([email protected]). 1 2 F.A. ALHARGAN

Laplace transform, Mittag-Leﬄer’s theorem, and the residue theorem. Such techniques provided concise and elegant solutions both in the x-domain and s-domain, revealing profound underlining connections between the Heaviside prime-counting function, the Dirac delta function, and the zeta function. Which I hope will provide another angle to address prime computations, primality testing, and prime factorization. 2. The Riemann Hypothesis. 2.1. Principle Zeros of the Zeta Function. For the case of the Riemann zeta function ζ(s), it has been shown, by Riemann [1], that the zeta function satisﬁes the following functional equation πs (2.1) ζ(s) = 2sπs−1 sin Γ(1 − s)ζ(1 − s), 2 where the symmetrical form of the functional equation is given as

s 1−s − 2 s 2 1−s (2.2) π Γ( 2 )ζ(s) = π Γ( 2 )ζ(1 − s).

We note that ζ(s) has zeros at s = sm = σm + itm, s =s ¯m = σm − itm, and s = −2m with m = 1, 2, 3,... . Many assume, from the functional equation (2.2) for ζ(1 − s), that s = 1−sm and s = 1−s¯m are also zeros of zeta. Nevertheless, the principle zeros of ζ(s) are determined only by using the pure argument s in ζ(s); hence, the principle zeros are only at s = sm, s =s ¯m, and s = −2m. Therefore, the sums and products of ζ(s) should only be over the zeros s = sm, s =s ¯m, and s = −2m, whenever appropriate, contrary to the usual statement that ”the infinite product is understood to be taken in an order which pairs each root ρ with the corresponding root 1 − ρ” [6] p.39. For clarity, I have rephrased the statement to ”the ζ(s) infinite product is understood to be taken in an order which pairs each root sm with the corresponding conjugate roots ¯m”; the difference is minor though the impact is tremendous. Now, the locations of the non-trivial zeros are determined by considering the Euler product of ζ(s) over the set of the prime numbers {2, 3, 5, . . . , pm,... }, given by

Y 1 (2.3) ζ(s) = , 1 − 1 p ps

which shows that ζ(s) does not have any zeros for <(s) > 1, and by the functional Equation (2.1), no zeros for <(s) < 0; save for the trivial zeros at s = −2m, due πs to the sin( 2 )Γ(1 − s) term. Jacques Hadamard (1896) [3] and Charles Jean de la Vallée-Poussin [11] independently proved that there are no zeros on the line <(s) = 1. In addition, considering the functional equation and the fact that there are no zeros with a real part greater than 1, it follows that all non-trivial zeros must lie in the interior of the critical strip 0 < <(s) < 1. Hardy and Littlewood (1921) [12] have 1 shown that there are infinitely many non-trivial zeros sm on the critical line s = 2 +it. We note that the non-trivial principle zeros of ζ(s) are located only in the strip 1 2 ≤ <(s) < 1, as shown in Figure (1), whereas the non-trivial zeros of ζ(1 − s) are 1 located in the strip 0 < <(s) ≤ 2 . Although this is a minor definition clarification, it is critical in proving the Riemann Hypothesis. This has been overlooked, as 1−sm =s ¯m for all the known zeros; thus, the product or sum over the zeros (1 − sm) is the same as the product or sum overs ¯m for the first ten trillion known zeros [13]. PROOF OF RIEMANN HYPOTHESIS 3

t The Critical Strip

ζ(s)

1 − s¯m sm

σ = 0 1 σ = 1 σ σ = 2

1 − sm s¯m ζ(1 − s)

Fig. 1. The Critical Strip.

2.2. Sums and Products for Zeta Function. In this section, the sum over the principle poles of a reciprocal function of zeta is developed based on Mittag- Leﬄer’s theorem, in order to showcase the linkage to the Hadamard product over the principle zeros of zeta, by considering a normalized function of ξ(s) given by

s s − 2 (2.4) f(s) = 2ξ(s) = ζ(s)(s − 1)sΓ( 2 )π , which is an entire function with f(s) = f(1 − s), f(1) = f(0) = 1, and has principle zeros only at s = sm and s =s ¯m. Thus, the ζ(s) inﬁnite product is understood to be taken in an order which pairs each root sm with the corresponding conjugate root s¯m. Now, taking the log, we have

s s (2.5) ln f(s) = ln 2 + ln ξ(s) = ln ζ(s) + ln(s − 1) + ln s + ln Γ( 2 ) − 2 ln π. Diﬀerentiating, we have

0 0 0 0 s f (s) ξ (s) ζ (s) 1 1 Γ ( 2 ) 1 (2.6) = = + + + s − 2 ln π, f(s) ξ(s) ζ(s) (s − 1) s Γ( 2 ) which gives

0 f (0) (2.7) = ln 2π − 1 − 1 γ − 1 ln π. f(0) 2 2

0 f (s) Note that f(s) has simple poles at the same zeros of ξ(s) (i.e., the poles are at s = sm and s =s ¯m). Now, using Mittag-Leﬄer’s theorem for the sum over the poles of the function 0 f (s) f(s) , we obtain

0 0 s ζ (s) 1 1 Γ ( 2 ) 1 + + + s =[ln 2π − 1 − 2 γ] ζ(s) (s − 1) s Γ( 2 ) (2.8) ∞ X 1 1 1 1 + + + + . (s − s ) s (s − s¯ ) s¯ m=1 m m m m 4 F.A. ALHARGAN

Integrating Equation (2.8) and taking the antilog, we have

∞ 1 s [ln 2π−1− γ]s Y s s s s (2.9) ζ(s)(s − 1)sΓ( ) = e 2 1 − e sm 1 − e s¯m , 2 s s¯ m=1 m m

1 which was proved by Hadamard [2]. Note the 2 ln π term canceled out, as it appears on both sides of the equation. Also, using Mittag-Leﬄer’s theorem for the following function

0 f (s) (2.10) F (s) = s =⇒ F (0) = 0, f(s) we have

0 0 ∞ ζ (s) 1 1 Γ ( s ) X 1 1 (2.11) + + + 2 − 1 ln π = + . ζ(s) (s − 1) s Γ( s ) 2 (s − s ) (s − s¯ ) 2 m=1 m m Integrating and taking the antilog, we have

∞ s Y s s (2.12) ζ(s)(s − 1)sΓ( s )π− 2 = 1 − 1 − ; 2 s s¯ m=1 m m that is,

∞ s Y s(2σm − s) (2.13) 2ξ(s) = ζ(s)(s − 1)sΓ( s )π− 2 = 1 − , 2 s s¯ m=1 m m

which was given by Riemann [1], in a logarithmic form with minor diﬀerence from the 1 modern deﬁnition of ξ(s). He set s = 2 + ti to obtain X tt (2.14) log ξ(t) = log 1 − + log ξ(0); αα that is,

Y tt (2.15) ξ(t) = ξ(0) 1 − . αα 2.3. A Proof of the Riemann Hypothesis. Theorem 2.1. The Riemann zeta function ζ(s) has only two independent sets of principle zeros, M and S. The set M of all principle trivial zeros of ζ(s) lies on the real negative axis with imaginary part t = 0, whereas the set S of all principle 1 non-trivial zeros of ζ(s) lies on the imaginary line with real part σ = 2 , as shown in Figure (2).

Proof. It has been shown, by Riemann [1], that the zeta function satisﬁes the following functional equation: πs (2.16) ζ(s) = 2sπs−1 sin Γ(1 − s)ζ(1 − s), 2 Now, if ζ(s) = 0, then from Equation (2.16), we have PROOF OF RIEMANN HYPOTHESIS 5

t sm

−2m 1 σ 2

s¯m

Fig. 2. ζ(s) Zeros Location.

πs (2.17) sin Γ(1 − s) = 0, 2 or

(2.18) ζ(s) = ζ(1 − s) = 0.

From Equation (2.17), we can obtain the set M of all trivial zeros of ζ(s) (i.e., M = {−2, −4,..., −2m, . . . }, where m is a positive integer) and, from Equation (2.18), we can obtain another independent set S of all non-trivial zeros of ζ(s), S = 1 {s1, s2, . . . , sm,... }, with sm = σm ± itm, where 2 ≤ σm < 1, tm are real numbers, and i is the imaginary unit. Now, by Equation (2.13), we have ∞ s Y s(2σm − s) (2.19) 2ξ(s) = ζ(s)(s − 1)sΓ( s )π− 2 = 1 − , 2 s s¯ m=1 m m and, considering the case of the limit when s → 1, we have ∞ 1 Y (2σm − 1) (2.20) lim[ζ(s)(s − 1)]Γ( 1 )π− 2 = 1 − . s→1 2 s s¯ m=1 m m It is well-known that

1 1 lim ζ(s)(s − 1) = 1 and Γ( ) = π 2 . s→1 2 Therefore, Equation (2.20) becomes ∞ Y (2σm − 1) (2.21) 1 = 1 − , s s¯ m=1 m m and since

1 (2.22) 2 ≤ σm < 1 for all the principle non-trivial zeros (sm = σm ± itm) of ζ(s), it implies that

(2.23) 0 ≤ (2σm − 1) < 1. 6 F.A. ALHARGAN

Therefore, Equation (2.21) is true only when (2σm − 1) = 0, which requires that 1 σm = 2 for all the non-trivial zeros of ζ(s). This concludes the proof of the Riemann Hypothesis that: the real part of every non-trivial zero of the Riemann zeta function 1 is σm = 2 . Also, the proof can be stated in a concise form as (2.24) ∞ s Y s(2σm − s) s − 2 1 ∵ ζ(s)(s − 1)sΓ( 2 )π = 1 − & 2 ≤ σm < 1 , sms¯m

lim m=1  s w

  1 = w s

→  w   w y1 y ∞ Y (2σm − 1) 1 1 = 1 − ==⇒ (2σ − 1) = 0 =⇒ σ = . ∴ s s¯ m m 2 m=1 m m

1 To validate the result, with σm = 2 , Equation (2.19) can be restated as

∞ s Y s(1 − s) (2.25) 2ξ(s) = ζ(s)(s − 1)sΓ( s )π− 2 = 1 − , 2 s s¯ m=1 m m

from which we see that the right hand side of Equation (2.25) is unchanged when s is replaced by (1 − s), obtaining the expressions for ζ(1 − s) and ξ(1 − s) as

∞ 1−s Y s(1 − s) (2.26) 2ξ(1 − s) = ζ(1 − s)s(s − 1)Γ( 1−s )π− 2 = 1 − . 2 s s¯ m=1 m m

Therefore, Equations (2.25) and (2.26) are equal, as validated by the well-known ξ(s) functional equation, given by

(2.27) ξ(s) = ξ(1 − s).

1 If any zero sm has σm 6= 2 in Equation (2.19), then it implies that ξ(s) 6= ξ(1 − s), 1 which would contradict Equation (2.27). Therefore, all σm must be equal to 2 . From this, we can hypothesize that the product form of the ξ(s) in Equation (2.14) developed by Riemann [1] was very likely to have been the source of inspiration for the Riemann Hypothesis. 3. The Prime-Counting Function. In this section, I will revisit the prime- counting function analysis. Recasting Riemann’s [1] synthesis and results in a format; that will consolidate diverse elements such as the Heaviside step function, the von Mangoldt function, the Dirac delta function, and the Chebyshev function. Then, I will employ the Laplace transform to obtain the prime-counting function in the s- domain. Also, I will utilize the residue theorem to drive the prime-counting function in terms of ζ(s) pole and zeros. Some of these results are already available in the literature in one form or another. However, here I will organize them logically and bridge some crucial gaps to demonstrate the underlining relationships. 3.1. The Prime-Counting Function in the x-domain. The number of primes less than a given magnitude x can be formulated in the x-domain on a fundamental building block, using the staircase Heaviside step function H(ln x − ln p) as a base for PROOF OF RIEMANN HYPOTHESIS 7

the prime counting function π(x), see Figure3, given by

X (3.1) π(x) = H(ln x − ln p), p

where p is the set of all prime numbers {2, 3, 5, ··· , pm, ···}.

π(x)

x 2 3 5 7

Fig. 3. Prime Counting Heaviside Step Function.

Also, in terms of prime harmonics pk, see Figure4, we have

1 X 1/k X (3.2) π(x k ) = H(ln x − ln p) = H(ln x − k ln p). p p

where k ∈ N.

1 π(x 2 )

x 2 3 4 5 7 9

1 Fig. 4. Prime Counting Function π(x k ) .

Diﬀerentiating the Heaviside function Equation (3.1); gives the Dirac delta function δ(x), see Figure5, thus we have

X 1 (3.3) π0(x) = δ(ln x − ln p). x p 8 F.A. ALHARGAN

x π0(x)

x 2 3 5 7

Fig. 5. Prime Dirac Delta Function δ(ln x − ln p).

Now, from Riemann’s deﬁnition of J(x) in terms of the prime counting function π(x), as

X 1 1 k (3.4) J(x) = k π(x ). k∈N and the inverse

X µ(k) 1 k (3.5) π(x) = k J(x ). k∈N where µ(k) is the M¨obiusfunction. Thus, from Equation (3.2) and Equation (3.4), we have

∞ X X X Λ(n) (3.6) J(x) = 1 H(ln x − k ln p) = H(ln x − ln n), k ln(n) k∈N p n=2 where the von Mangoldt function, denoted by Λ(n), is deﬁned as ( ln p if n = pk for some prime p and integer k ≥ 1, (3.7) Λ(n) = 0 otherwise.

Diﬀerentiating Equation (3.6), we have

0 X 1 0 1 X 1 X 1 J (x) = π (x k ) = δ(ln x − k ln p) k k x k∈N k∈N p (3.8) ∞ ∞ X Λ(n) X Λ(n) = δ(ln x − ln n) = δ(ln x − ln n). x ln(n) x ln(x) n=2 n=2 Furthermore, the ﬁrst Chebyshev function is deﬁned by X (3.9) ϑ(x) = ln p, p≤x PROOF OF RIEMANN HYPOTHESIS 9

and based on Heaviside function can be deﬁned as X (3.10) ϑ(x) = H(ln x − ln p) ln p, p or in a prime factorization format, as Y (3.11) eϑ(x) = pH(ln x−ln p). p

Diﬀerentiating Equation (3.10), we have

X 1 (3.12) ϑ0(x) = δ(ln x − ln p) ln p. x p It is interesting to note that the relation between the differential of the prime-counting function in Equation (3.3) and the differential of the first Chebyshev function in Equation (3.12), can be stated as

(3.13) ϑ0(x) = π0(x) ln x.

Similarly, the second Chebyshev function ψ(x) is deﬁned with the sum extending over all prime powers not exceeding x, as X X X (3.14) ψ(x) = ln p = Λ(n), k∈N pk≤x n≤x or based on the Heaviside function, as

∞ X X X (3.15) ψ(x) = H(ln x − k ln p) ln p = Λ(n) H(ln x − ln n), k∈N p n=2

which can be represented in the following fascinating factorization format Y Y (3.16) eψ(x) = pH(ln x−k ln p). k∈N p Diﬀerentiating Equation (3.15), we have

∞ X X ln p X Λ(n) (3.17) ψ0(x) = δ(ln x − k ln p) = δ(ln x − ln n). x x k∈N p n=2 From Equations (3.8) and (3.17), it is observed that

∞ X X X (3.18) xψ0(x) = xJ 0(x) ln x = ln p δ(ln x − k ln p) = Λ(n)δ(ln x − ln n). k∈N p n=2

Also, we observe that Equation (3.18) is basically a delta function, given by

( ln p if x = pk a prime with an integer k ≥ 1, (3.19) xψ0(x) = xJ 0(x) ln x = 0 otherwise. 10 F.A. ALHARGAN

Now, multiplying Equation (3.2) by x−s−1 and integrating, we have

∞ ∞ Z Z 1 −s−1 X −s−1 (3.20) π(x k ) x dx = H(ln x − k ln p) x dx,

1 1 p integrating the right hand-side of Equation (3.20) by parts, we have

∞ Z 1 −s−1 1 X −sk (3.21) π(x k ) x dx = p . s 1 p

Thus, from Equation (3.21) and Equation (3.4), we have

∞ Z 1 X X (3.22) J(x) x−s−1 dx = 1 p−sk. s k 1 k∈N p Here, we observe that the right hand-side of Equation (3.22) is basically the log of the Euler product of ζ(s). Therefore, Equation (3.22) can be restated as

∞ Z ln ζ(s) (3.23) J(x)x−s−1 dx = , (< s > 1). s 1

Equation (3.23) was one of the main results in Riemann’s paper [1]. However, the above analysis reveals the profound direct connection between the Heaviside prime- counting function and the zeta function. This approach shifts the perspective to a new paradigm from number theory to signal processing theory using Riemann spectrum [14], which will enable us to exert the signal processing arsenal to tackle some prime numbers enigmas. 3.2. The Prime-Counting Function in the s-domain. The Fourier analysis has been the mainstay in the literature for tackling the connection between zeta and the prime number counting function. However, the Laplace transform is a generalized Fourier transform that provides elegant and concise solutions; linking the s-domain with the x-domain. The analysis so far has been based on the x-domain. Here, I will formulate the functions in the s-domain, then demonstrate their links via the Laplace transform by ﬁrst employing the well-known Laplace transforms for the Heaviside and Dirac delta functions, given by e−sk ln n (3.24) L{H(ln x − k ln n)} = , s and 1 (3.25) L δ(ln x − k ln n) = e−sk ln n. x Thus, the Laplace transform of the prime counting function Equation (3.1), from π(x) to Π(s), is obtained as

X X e−s ln p (3.26) Π(s) = L{π(x)} = L{H(ln x − ln p)} = , s p p PROOF OF RIEMANN HYPOTHESIS 11 and X 1 X (3.27) sΠ(s) = L{π0(x)} = L δ(ln x − ln p) = e−s ln p. x p p Here, it is important to note the definition of the prime-counting function in the s- domain, denoted by the symbol Π(s). This is a new definition and should not be confused with the same symbol used in the literature for different purposes. Thus, we can define the s-domain prime-counting function as

1 X 1 (3.28) Π(s) := , (< s > 1). s ps p

The deﬁnition in Equation (3.28) is the base prime numbers sub-sum of the ζ(s) function. It contains only the base primes; i.e. excluding prime number harmonics and multiples of primes from the sum. Now, we can utilize Equation (3.24) to obtain the Laplace transform of Equations (3.6), (3.10) and (3.15), as

∞ X X X Λ(n) (3.29) sJ(s) = 1 e−ks ln p = e−s ln n, k ln(n) k∈N p n=2

X (3.30) sΘ(s) = ln p e−s ln p, p and

∞ X X X (3.31) sΨ(s) = ln p e−ks ln p = Λ(n)e−s ln n. k∈N p n=2 Here, we recall the log expansion of the Euler product of Riemann zeta function, which is given by

∞ X X X X Λ(n) 1 (3.32) ln ζ(s) = − ln(1 − e−s ln p) = 1 e−ks ln p = , k ln(n) ns p k∈N p n=2 and diﬀerentiating, we have

∞ ζ0(s) X ln p X X X (3.33) = = − ln p e−ks ln p = − Λ(n)e−s ln n. ζ(s) (1 − ps) p k∈N p n=2

Therefore, from Equations (3.28), (3.29), (3.30), (3.31)(3.32) and (3.33), we discover that in the s-domain, the relationship amongst the functions Π(s), ζ(s), J(s), Θ(s) and Ψ(s), are as follows

∞ X X X Λ(n) X (3.34) sJ(s) = ln ζ(s) = 1 e−ks ln p = e−s ln n = s Π(ks), k ln(n) k∈N p n=2 k∈N 12 F.A. ALHARGAN and

∞ ζ0(s) X X X X (3.35) sΨ(s) = − = ln p e−ks ln p = Λ(n)e−s ln n = s Θ(ks). ζ(s) k∈N p n=2 k∈N

Here, we observe the power of employing the Heaviside function and the s-domain analysis, which immediately demonstrate the profound relationship between ζ(s) and the prime counting function, for Equation (3.34) reveals that ln ζ(s) is the sum of all the harmonics of the prime counting function Π(s) in the s-domain. Remark 3.1. Here, it is important to highlight that the functions with these symbols: Π(s), J(s), Θ(s) and Ψ(s). That, I have newly defined in this paper, as the s-domain manifestation of their x-domain representation: π(x), J(x), ϑ(x) and ψ(x) respectively. These functions have not been defined previously in the literature. Thus should not be confused with any similar symbols you may encounter in the literature. 3.3. The Inverse Laplace Transform. Now, to obtain the functions in the x-domain, we employ the residue theorem to evaluate the inverse Laplace transform of the expressions; i.e. X (3.36) L−1 {F (s)} = Res [F (s)esy] , all poles where in our case y = ln x; and it is critical to take care of the effect of the term ln x 1 when differentiating, as the factor of x needs to be taken into account. Also, by Mittag-Leffler’s theorem, we have

∞ ζ0(s) 1 X 1 1 1 (3.37) = ln 2 − + + + , ζ(s) (s − 1) (s − s ) (s − s¯ ) (s + 2m) m=1 m m

ζ0(s) where the poles of the function ζ(s) are at s = 1, s = sm, s =s ¯m and s = −2m. Thus, the inverse Laplace transform, is obtained as follows

ζ0(s) X ζ0(s)es ln x xψ0(x) =L−1 {sΨ(s)} = L−1 − = − Res ; (3.38) ζ(s) ζ(s) all poles i.e.

∞ X (3.39) xψ0(x) = eln x − esm ln x + es¯m ln x + e−2m ln x. m=1

Also,

ζ0(s) X ζ0(s)es ln x (3.40) ψ(x) = L−1 {Ψ(s)} = L−1 − = − Res ; ζ(s) s ζ(s)s all poles i.e.

∞ X esm ln x es¯m ln x e−2m ln x (3.41) ψ(x) = eln x − ln 2π − + − . s s¯ 2m m=1 m m PROOF OF RIEMANN HYPOTHESIS 13

We note that in Equation (3.41), the terms 1 , 1 and 1 are due to 1 , and the sm s¯m 2m s ζ0(0) term ζ(0) = ln 2π is due to the pole at s = 0. Now, from Equations (3.18), (3.25), (3.33) and (3.39), we see that (3.42) ∞ ∞ X X xψ0(x) = xJ 0(x) ln x = x − xsm + xs¯m + x−2m = Λ(n)δ(ln x − ln n). m=1 n=2

Also, from Equations (3.15), (3.24), (3.33) and (3.41), we see that

∞ ∞ X xsm xs¯m x−2m X (3.43) ψ(x) = x − ln 2π − + − = Λ(n)H(ln x − ln n). s s¯ 2m m=1 m m n=2

It is interesting to observe that diﬀerentiating Equation (3.43) results in Equation (3.42), which validates the analysis. 3.4. The J(x) Function. Rearranging Equation (3.42), we have

∞ 1 X xsm−1 xs¯m−1 x−2m−1 (3.44) J 0(x) = − + + , ln x ln x ln x ln x m=1 or as stated by Riemann [1] for an approximate expression for the density of the prime numbers

∞ − 1 1 X x 2 cos(tm ln x) (3.45) J 0(x) = − 2 . ln x ln x m=1 Now, integrating Equation (3.44), results in the logarithmic integral function Li(x), giving

∞ X (3.46) J(x) = Li(x) − Li(xsm ) + Li(xs¯m ) + Li(x−2m). m=1 Then the prime counting function is ﬁnally obtained as

∞ X µ(k) 1 X µ(k) X k sm/k s¯m/k −2m/k (3.47) π(x) = k Li(x ) − k Li(x ) + Li(x ) + Li(x ). k∈N k∈N m=1

Equation (3.47) is exactly the core result of Riemann’s paper [1]. Furthermore, the above analysis demonstrate clear insight into the relation of the prime counting function and the zeta function; as can be observed that the ﬁrst part of Equation (3.47) is due to the pole of ζ(s) at s = 1, and the second part is due to the zeros of ζ(s) at s = sm, s =s ¯m and s = −2m. Moreover, we observe that the term due to the pole is the major component of π(x), whereas the terms due to the real zeros are negligible. In contrast, the terms due to the complex zeros are the source of the sawtooth-like wave component, which will be illustrated later. The sums are conditionally convergent with a slow convergence rate. Although Equation (3.47) was a landmark result that laid the foundations for prime numbers analysis, it is cumbersome and not convenient for analyzing prime numbers. In the next section, I will develop more convenient expressions. 14 F.A. ALHARGAN

3.5. Chebyshev ψ(x) Function. Utilizing the proof of the Riemann Hypothe- 1 sis, that the non-trivial zeros of zeta have real part equal to 2 . i.e. the zeros have a 1 format of sm = 2 + i tm, and noting that

∞ X x−2 (3.48) x−2m = . 1 − x−2 m=1 Thus, Equation (3.42) can be expressed as

−3 ∞ ∞ 0 x − 1 X X Λ(n) (3.49) ψ (x) = 1 − − 2x 2 cos (t ln x) = δ(ln x − ln n). (1 − x−2) m x m=1 n=2

Also, integrating Equation (3.49) or rearranging Equation (3.43), we have (3.50) ∞ 1 1 1 X [cos(tm ln x) + 2 tm sin(tm ln x)] ψ(x) = x − ln 2π − ln(1 − ) − 4x 2 . 2 x2 4 t2 + 1 m=1 m

Note that Equation (3.50), was proved in 1895, by Hans Carl Friedrich von Man- goldt ([15] p. 294 Equ.58), and was stated in the paper as

∞ 1 1 1 X [cos(αν ln x) + 2 αν sin(αν ln x)] (3.51) Λ(x, 0) = x − ln 2π − ln(1 − ) − x 2 . 2 x2 1 + α2 ν=1 4 ν where he showed that Λ(x, r) ([15] p. 279 Equ.38), is given by

a+ih 1 Z ζ0(s + r) xs (3.52) − lim · · ds = Λ(x, r). h→∞ 2πi ζ(s + r) s a−ih

ζ0(s) Equation (3.52) is basically the inverse Laplace transform of ζ(s) s when r = 0, from which we see that ψ(x) = Λ(x, 0). Of course, some of these results are not new, already Edwards ([6] p.50) had shown a short method to obtain Equation (3.51). However, in this paper, I have employed the residue theorem to prove the results within few steps. Furthermore, I have demonstrated via Laplace transform the links of the Chebyshev ψ(x) function and its derivative to the prime counting functions, the Heaviside function, and the Dirac delta function, which provides a new perspective and simpliﬁes the analysis. Now, the term [ln(1 − x−2)] has a negligible value, thus Equations (3.49) and (3.50) can be approximated to

∞ ∞ 0 1 X X (3.53) xψ (x) = x − 2x 2 cos (tm ln x) = Λ(n)δ(ln x − ln n). m=1 n=2

and (3.54) ∞ ∞ 1 X cos(tm ln x) sin(tm ln x) X ψ(x) x − ln 2π − x 2 + = Λ(n)H(ln x − ln n). u t2 2t m=1 m m n=2 PROOF OF RIEMANN HYPOTHESIS 15

Equations (3.53) and (3.54) are in essence contain the Riemann spectrum of ζ(s) 1 non-trivial zeros, i.e. the set { 2 + itm}. These equations are powerful in locating the prime numbers and their harmonics; for Equation (3.53) is essentially a delta function of the prime numbers harmonics, whereas Equation (3.54) is a Heaviside staircase function of the prime numbers harmonics. Comparing Equation (3.33) and Equation (3.53), we observe that in the s-domain the function Ψ(s) is summed over the prime numbers; and has poles at the zeros of 0 zeta at s = sm, s =s ¯m and s = −2m. In contrast, the x-domain function ψ (x) is summed over the zeros of zeta; and has poles at the harmonics of the prime numbers.

Fig. 6. x ψ0(x) with x continuous.

Although Equations (3.53) and (3.54) are slow and conditionally convergent, the two equations provide better speed and accuracy for the primality test than current algorithms. In ﬁgures (6) and (7); with the computation executed at steps of x = 0.01, we can see the harmonics and Gibbs phenomenon. Also, we observe the prime locations at 101, 103, 107, 109; the harmonic primes at 121 = 112, 125 = 53, 128 = 27, and the larger primes at 127, 131, 137, 139; whereas the rest of the natural non- prime harmonics integers are small. In Figure (7), the Heaviside staircase prime harmonics counting function is observed. Also, we can observe the sawtooth-like waveform component of ψ(x) in Figure (8). 16 F.A. ALHARGAN

Fig. 7. ψ(x) with x continuous.

Fig. 8. Sawtooth-like waveform component of ψ(x) .

4. The Zero-Counting Function. In this section, and in light of the proof of the Riemann Hypothesis, I will revisit the zero-counting function analysis. Then I will reformulate the equations to obtain a very accurate zero-counting function. Now, in the range {0,T }, the number of roots of ξ(s); was conjectured by Riemann [1], as approximately T T T (4.1) = 2π ln 2π − 2π , PROOF OF RIEMANN HYPOTHESIS 17 and some 46 years later was proved by H. von Mangoldt [16], the prove was outlined by Ivic ([7], p. 17), where he showed that the number of zeros is given approximately by Z 0 T T T 7 1 ζ (s) (4.2) N(T ) = 2π ln 2π − 2π + 8 + π = ds, L ζ(s) and demonstrated that Z ζ0(s) (4.3) = ds = O(ln T ). L ζ(s) Although the integral in Equation (4.3) is small compared to the major elements in Equation (4.2), it still contains the sawtooth-like waveform component, that I will demonstrate shortly. Now, recalling Riemann [1] main justification of Equation (4.1), quoted as follows: ”because the integral R d log ξ(t), taken in a positive sense around the region consisting of the values of t whose imaginary parts lie between 1 1 2 i and − 2 i and whose real parts lie between 0 and T , is (up to a frac- 1 T tion of the order of magnitude of the quantity T ) equal to (T log 2π − T )i; this integral however is equal to the number of roots of ξ(t) = 0 lying within this region, multiplied by 2πi. One now finds indeed approximately this number of real roots within these limits, and it is very probable that all roots are real.” In essence, Riemann instinctively was invoking Cauchy’s argument principle, for ξ(s) is a meromorphic function inside and on some closed contour D, and ξ(s) has no zeros or poles on D, thus 1 I ξ0(s) (4.4) ds = Z − P, 2πi D ξ(s) where Z and P denote the number of zeros and poles of ξ(s); inside the contour D. Now, from the proof of the Riemann Hypothesis, which implies that ξ(s) has 1 simple zeros only on the critical line <(s) = 2 , at s = sm and s =s ¯m. Then, we can define the zero counting function ν(t) of the number of zeros of ξ(s), in the range {s, s¯}, as I 0 I 0 0 s ξ (s) ζ (s) 1 1 Γ ( 2 ) 1 (4.5) 4πiν(t) = ds = + + + s − ln π ds. D ξ(s) D ζ(s) (s − 1) s Γ( 2 ) 2 where the closed contour D encompasses the critical strip [0 ≤ <(s) ≤ 1]. 0 0 s s Now, since all the poles of [ζ (s)/ζ(s)+Γ ( 2 )/Γ( 2 )] are on the critical line <(s) = 1 2 , and the contour L1 encloses all these poles in the range froms ¯ to s, whereas the contour L2 encloses only the two poles s = 0 and s = 1, see Figure (9). Then we can rearrange Equation (4.5), as I ζ0(s) Γ0( s ) 1 I 1 1 (4.6) 4πiν(t) = + 2 − ln π ds + + ds, ζ(s) Γ( s ) 2 (s − 1) s L1 2 L2

The contour integration around L1, can be transformed to a line integral, as follows I Z s+ Z s− Z s¯− Z s¯+ Z s (4.7) = lim + − − = 2 →0 L1 s¯+ s+ s− s¯− s¯ 18 F.A. ALHARGAN

D L1 sm

1 σ 2

s¯m

Fig. 9. ζ(s) Critical Strip, Contours D, L1 and L2.

and the contour integration around L2, is given by I 1 1 (4.8) + ds = 4πi, L2 (s − 1) s thus, we have

Z s 0 0 s ζ (s) Γ ( 2 ) 1 (4.9) 4πiν(t) = 2 + s − ln π ds + 4πi, s¯ ζ(s) Γ( 2 ) 2 integrating, we ﬁnally have

s s¯ s s¯ − 2 − 2 (4.10) 2πiν(t) = ln ζ(s) − ln ζ(¯s) + ln Γ( 2 ) − ln Γ( 2 ) + ln π − ln π + 2πi, or

s −it 2πiν(t) [ζ(s)Γ( 2 )π ] 2πi (4.11) e = s¯ e ; [ζ(¯s)Γ( 2 )] i.e. (4.12) 1 1 1 t 1 t 2πiν(t) = ln ζ( 2 + it) − ln ζ( 2 − it) + ln Γ( 4 + i 2 ) − ln Γ( 4 − i 2 ) − it ln π + 2πi. Diﬀerentiating Equation (4.12), we have

0 1 0 1 0 1 t 0 1 t 0 ζ ( 2 + it) ζ ( 2 − it) Γ ( 4 + i 2 ) Γ ( 4 − i 2 ) (4.13) 2πiν (t) = 1 − 1 + 1 t − 1 t − i ln π. ζ( 2 + it) ζ( 2 − it) Γ( 4 + i 2 ) Γ( 4 − i 2 ) Utilizing Stirling approximation √ z −z − 1 (4.14) Γ(z) ∼ (z) e z 2 2π, PROOF OF RIEMANN HYPOTHESIS 19 and

Γ0(z) 1 (4.15) ∼ ln z − , Γ(z) 2z we have

s s¯ s s 1 1 s¯ s¯ 1 1 (4.16) ln Γ( 2 ) − ln Γ( 2 ) ∼ 2 ln 2 − 2 s − 2 ln s − 2 ln 2 + 2 s¯ + 2 lns, ¯ or

s s¯ s s s¯ s¯ 1 1 (4.17) ln Γ( 2 ) − ln Γ( 2 ) ∼ 2 ln 2 − 2 ln 2 + 2 lns ¯ − 2 ln s − it.

Thus, ﬁnally we have

1 t 1 t 1 t 1 t 2πiν(t) =( 4 + i 2 ) ln( 4 + i 2 ) − ( 4 − i 2 ) ln( 4 − i 2 ) + 1 ln( 1 − i t ) − 1 ln( 1 + i t ) (4.18) 2 4 2 2 4 2 1 1 X + ln ζ( 2 + it) − ln ζ( 2 − it) − it ln πe = 2πi H(t − tm), m and from Equation (4.13), we have

0 1 1 1 1 2πiν (t) = ln( 2 + it) − 1 − ln( 2 − it) + 1 ( 2 + it) ( 2 − it) (4.19) ζ0( 1 + it) ζ0( 1 − it) X + 2 − 2 − i ln π = 2πi δ(t − t ). ζ( 1 + it) ζ( 1 − it) m 2 2 m

Equation (4.18) is a very accurate approximation, and it is a sum of diﬀerences between complex numbers and their conjugates, thus the result will always be imaginary number as expected. Further approximation of the log part gives

t t 7 1 1 1 (4.20) ν(t) = 2π ln 2eπ + 8 + 2πi ln ζ( 2 + it) − ln ζ( 2 − it) .

1 1 Figure (10), shows the sawtooth-like waveform component ln ζ( 2 + it) − ln ζ( 2 − it) of ν(t). It is observed that the component magnitude is less than one. However, it has a vital contribution to the accuracy of the zero-counting function; which turns it into a Heaviside staircase step function, as shown in Figure (11). 20 F.A. ALHARGAN

Fig. 10. The sawtooth-like waveform component of ν(t).

Fig. 11. ν(t) vs. N(t).

Now, from equation (3.34), we have

1 1 X 1 1 1 1 1 1 (4.21) ln ζ( 2 + it) − ln ζ( 2 − it) = k ( 2 + it)Π(k( 2 + it)) − k ( 2 − it)Π(k( 2 − it)), k PROOF OF RIEMANN HYPOTHESIS 21 or

X X k 1 1 1 − 2 (4.22) ln ζ( 2 + it) − ln ζ( 2 − it) = 2i k p sin(tk ln p), p k giving

X X k t t 7 1 1 − 2 (4.23) ν(t) = 2π ln 2eπ + 8 + π k p sin(tk ln p). p k

We observe from Equation (4.21), the direct relation between the zero-counting function ν(t) and the s-domain prime-counting function Π(s). Furthermore, we observe in Equation (4.23), the direct relationship between the number of ζ(s) zeros and the prime harmonics. Although the equation is slow for computational purposes, it reveals the underlining relationship between the zero-counting function and the primes. Furthermore, it exposes the source of the sawtooth-like waveform effect as the spectrum sum of the prime harmonics. Finally, Figure (11) shows comparisons between ν(t) and N(t), and it confirms that the zero counting function is a Heaviside staircase step function, and its differential ν0(t) is an impulse Dirac delta function; as can be seen in Figure (12).

Fig. 12. |ν0(t)| .

5. Recommendations for Future Research. Here I will summarize some of the key results and relations, then I will outline some interesting topics, that I think are worthy of further research and exploration. 5.1. Natural Number Counting Function. Now, consider the natural number- counting function φ(x) based on Heaviside function, which we can deﬁned as X (5.1) φ(x) = H(ln x − ln n), n 22 F.A. ALHARGAN

and X 1 (5.2) φ0(x) = δ(ln x − ln n). x n Taking the Laplace transform of the above equations, we have

∞ X 1 X (5.3) L{φ0(x)} = L δ(ln x − ln n) = e−s ln n = ζ(s), x n n=1 and

∞ X X e−s ln n (5.4) L{φ(x)} = L{H(ln x − ln n)} = = 1 ζ(s). s s n n=1

1 Therefore, the inverse Laplace transform of s ζ(s) to the x-domain, manifests itself as the natural number counting function φ(x). Also, we observe from Equation (5.3) an interesting x-domain representation of the inverse Laplace transform of ζ(s) as a decreasing Dirac impulse function. Equation (5.4) can be utilized to count many variations of numbers, such as the numbers of the form nr, exploring the features of this equation is an interesting topic to research. 5.2. The Prime Counting Function. Recalling the deﬁnition of the prime counting function in the x-domain, as X (5.5) π(x) = H(ln x − ln p), p with

X 1 1 k (5.6) J(x) = k π(x ), k∈N and the reverse

X µ(k) 1 k (5.7) π(x) = k J(x ), k∈N where µ(k) is the M¨obiusfunction. Also, recalling the s-domain form, given by

X 1 (5.8) sΠ(s) = , < s > 1. ps p

with the relation to the zeta function, given by X (5.9) sJ(s) = ln ζ(s) = s Π(ks), k∈N and the reverse, as

X µ(k) X µ(k) (5.10) sΠ(s) = s J(ks) = ln ζ(ks). k k k∈N k∈N PROOF OF RIEMANN HYPOTHESIS 23

1 Fig. 13. <{sΠ(s)} along the critical line s = 2 + it.

Equation (5.8) for Π(s) exhibits elegance as well as deceptive simplicity, however, its complexity is revealed by Equation (5.10). The function Π(s) has poles at s = 0, s = 1, and at the zeros of ζ(s). Figure (13) shows the real part of sΠ(s); along the 1 1 critical line s = 2 , we also observe the poles at s = 2 + itm. Now, the inverse Laplace of Π(s) gives directly the prime-counting function π(x), i.e.

(5.11) π(x) = L−1 {Π(s)} .

From the Laplace transform properties, multiplication by s results in diﬀerentiation of π(x), and multiplication by −x results in diﬀerentiation of sΠ(s), thus we have

(5.12) − x π0(x) ln x = L−1 {[sΠ(s)]0} ; i.e. ( ) X µ(k) ζ0(ks) (5.13) x π0(x) ln x = L−1 − , k ζ(ks) k∈N Then using the residue theorem to evaluate the inverse Laplace, we have " # X X µ(k) ζ0(ks)es ln x (5.14) x π0(x) ln x = − Res , k ζ(ks) all poles k∈N giving

∞ X µ(k) 1 X X µ(k) sm s¯m 2m 0 k k k − k (5.15) xπ (x) ln x = k x − k x + x + x . k∈N m=1 k∈N 24 F.A. ALHARGAN

Integrating, we have

∞ X µ(k) 1 X µ(k) X k sm/k s¯m/k −2m/k (5.16) π(x) = k Li(x ) − k Li(x ) + Li(x ) + Li(x ). k∈N k∈N m=1

Here we come back a full circle to the same result. However, this approach gives much better clarity and coherence with a far fewer steps. The elegant s-domain forms present a new perspective in the relation between the ζ(s) function and the prime counting function Π(s). The behaviour of Π(s) needs further investigation that might reveal new insights into the computations of the primes. 5.3. The Chebyshev Function. Recalling the deﬁnition of the ﬁrst kind Cheby- shev function in the x-domain, as X (5.17) ϑ(x) = H(ln x − ln p) ln p, p

or in a factorization form, as Y (5.18) eϑ(x) = pH(ln x−ln p). p

Also, the ﬁrst kind Chebyshev function in the s-domain, as

X 1 (5.19) sΘ(s) = ps ln p, < s > 1. p

Y 1 (5.20) sΘ(s) = ln p ps , < s > 1. p

Moreover, its relation to the zeta function is given by

ζ0(s) X (5.21) sΨ(s) = − = s Θ(ks). ζ(s) k∈N and the reverse, as

X µ(k) X µ(k) ζ0(ks) (5.22) sΘ(s) = s Ψ(ks) = − . k k ζ(ks) k∈N k∈N

In fact, the prime counting function π(x) and the ﬁrst Chebyshev function ϑ(x), can be directly evaluated from Equation (5.22), using the inverse Laplace transform. We note that the function Θ(s) has poles at s = 0, s = 1, and the zeros of ζ(s). This again, provides another perspective in the relation between ζ(s) and the Chebyshev function expressed in the s-domain. The behaviour of Θ(s) needs further investigation, that could yield greater insights into the prime factorization. PROOF OF RIEMANN HYPOTHESIS 25

6. Conclusions. Proof of the Riemann Hypothesis would unravel many of the mysteries surrounding the distribution of prime numbers, which are at the heart of all encryption systems. In addition, proof of the Riemann Hypothesis would, as a consequence, prove many of the propositions known to be true under the Riemann Hypothesis. The proof demonstrated in this paper was based on a basic insight into the product expansion of the Riemann zeta function, as available from Hadamard’s publication in 1893 and Riemann’s publication in 1859, as well as clarifying that the product expansion is only over the principle non-trivial zeros of zeta. Sometimes, the truth is hidden in plain sight. Furthermore, in this paper, I have demonstrated several techniques with a new perspective. These included the Heaviside function, Dirac delta function, the Laplace transform, Mittag-Leffler’s theorem, and the residue theorem. The techniques have provided concise and elegant solutions; that immediately revealed the profound underlining connection between the Heaviside prime-counting function, the Dirac delta function, and the zeta function. This novel approach shifts the perspective to a new paradigm, from number theory to signal processing theory. These results will enable us to exert the signal processing arsenal to tackle some prime numbers enigmas. The new defined s-domain prime-counting function Π(s) and Chebyshev function Ψ(s) provide another angle to address prime computations, primality testing, and prime factorization. Acknowledgements. I would like to thank Dr. Sami M. Alhumaidi director general of PSDSARC, Prof. Mohammed I. Alsuwaiyel, and Dr. Fawzi A. Al-thukair for their feedback on the first draft and their encouragement. Naturally, All Errors are My Own.

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