Generalized Riemann Hypothesis Léo Agélas

Home , Conjecture, Dirichlet character, Prime zeta function, Riemann hypothesis

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LéoAgélas Department of Mathematics, IFP Energies nouvelles, 1-4, avenue de Bois-Préau,F-92852 Rueil-Malmaison, France

Abstract (Generalized) Riemann Hypothesis (that all non-trivial zeros of the (Dirichlet L-function) zeta function have real part one-half) is arguably the most important unsolved problem in contemporary mathematics due to its deep relation to the fundamental building blocks of the integers, the primes. The proof of the Riemann hypothesis will immediately verify a slew of dependent theorems (Borwien et al.(2008), Sabbagh(2002)). In this paper, we give a proof of Gen- eralized Riemann Hypothesis which implies the proof of Riemann Hypothesis and Goldbach’s weak conjecture (also known as the odd Goldbach conjecture) one of the oldest and best-known unsolved problems in number theory.

1. Introduction

The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. The Riemann zeta function is deﬁned (Abramowitz and Stegun(1964) p. 807) by the series, ∞ X 1 ζ(s) = , s ∈ , (1) ns C n=1 which is analytic in <(s) > 1 (see Borwien et al.(2008)). The ﬁrst connection between zeta functions and prime numbers was made by Euler when he showed for s real the following beautiful identity (see Stein and Shakarchi(2003),Bor- wien et al.(2008),Conrey(2003),Sabbagh(2002)):

−1 Y 1 ζ(s) = 1 − , s ∈ ; <(s) > 1, (2) ps C p∈P

def where P = {2, 3, 5, 7, 11, 13, ...} is the set of prime positive integers p. On the other side, Riemann proved that ζ(s) has an analytic continuation to the whole complex plane except for a simple pole at s = 1 (see Riemann(1859),

Email address: [email protected] (L´eoAg´elas)

Preprint March 4, 2019 H.M. Edwards(1974)). Moreover, he showed that ζ(s) satisﬁes the functional equation (see Titchmarsh(1986),H.M. Edwards(1974),Borwien et al.(2008)), πs ζ(s) = 2sπs−1 sin( )Γ(1 − s)ζ(1 − s), (3) 2 where Γ(s) is the complex gamma function. It was also shown that (see Titch- marsh(1986))

∗ 1. ζ(s) is nonzero in <(s) < 0, except for the real zeros {−2m}m∈N ,

∗ 2. {−2m}m∈N are the only real zeros of ζ(s) called trivial zeros, In 1859, B. Riemann formulated the following conjecture: Conjecture 1.1 (Riemann Hypothesis). All non trivial zeros of ζ(s) lies exactly 1 on <(s) = . 2 We know that the zeta function was introduced as an analytic tool for study- ing prime numbers and some of the most important applications of the zeta functions belong to prime number theory. Indeed, it was shown independently in 1896 by Hadamard and de la Vallée-Poussin, that ζ(s) has no zeros on <(s) = 1 (see Titchmarsh(1986) p. 45), which provided the first proof of the Prime Number Theorem: x π(x) ∼ (x → +∞), (4) log x where π(x) def= { number of primes p for which p ≤ x} (where x > 0). Their proof comes with an explicit error estimate: they showed in fact (see for example Theorem 6.9 in Montgomery and Vaughan(2006)), x π(x) = li(x) + O √ , (5) exp(c log x) uniformly for x ≥ 2. Here li(x) is the logarithmic integral, Z x dt li(x) def= . 2 log t Later, Von Koch proved that the Riemann hypothesis is equivalent to the ”best possible” bound for the error of the Prime Number Theorem (see Koch(1901)), namely Riemann hypothesis is equivalent to, √ π(x) = li(x) + O x log(x) . (6) The Riemann zeta function ζ(s) and the Riemann Hypothesis have been the object of a lot of generalizations and there is a growing literature in this regard comparable with that of the classical zeta function itself. The most direct gen- eralization, which is also what we will mainly deal with, concerns the Dirichlet L-functions with the corresponding Generalized Riemann Hypothesis. Dirichlet defined his L-functions in 1837 as follows. A function χ : Z 7−→ C is called a Dirichlet character modulo k if it satisfies the following criteria:

2 (i) χ(n) 6= 0 if (n, k) = 1; (ii) χ(n) = 0 if (n, k) > 1; (iii) χ is periodic with period k : that is χ(n + k) = χ(n) for all n; (iv) χ is (completely) multiplicative : that is χ(mn) = χ(m)χ(n) for all integers m and n.

The principal character (or trivial character) is the one such that χ0(n) = 1 whenever (n, k) = 1. Then, one can deﬁne the Dirichlet series for <(s) > 1,

∞ X χ(n) L(s, χ) = . (7) ns n=1 L(χ, s) can be analytically continued to meromorphic functions in the whole complex plane (see Theorem 10.2.14 in Cohen(2007)). If χ : Z 7−→ C∗ is a principal character, then L(s, χ) has a simple pole at s = 1 and is analytic everywhere, otherwise L(s, χ) is analytic everywhere (see Theorem 12.5 in Apostol (1976), see also Theorem 10.2.14 in Cohen(2007)). As in the case of the Riemann zeta function, by multiplicativity, there is an Eu- ler product decomposition over the primes, for <(s) > 1 (see Davenport(1980), Ellison and F. Ellison(1985), Serre(1986)),

−1 Y χ(p) L(s, χ) = 1 − . (8) ps p∈P Thanks to (8), we get (see Cohen(2007)[Corollary 10.2.15])

L(χ, s) 6= 0 for all s ∈ C; <(s) > 1. (9) For any Dirichlet character χ mod k there is a smallest divisor k0|k such that χ agrees with a Dirichlet character χ0 mod k0 on integers coprime with k. The resulting χ0 is called primitive and has many distinguished properties. First of all, χ being induced from χ0 means analytically that Y L(s, χ) = L(s, χ0) (1 − χ0(p)p−s), p|k whence L(s, χ) and L(s, χ0) have the same zeros in the critical strip 0 ≤ <(s) ≤ 1. Zeros outside this strip are well understood, indeed L(s, χ) 6= 0 if <(s) > 1 and for a primitive character χ, the only zeros of L(s, χ) for <(s) < 0 are as follows s = ε − 2m, ε ∈ {0, 1} such that χ(−1) = (−1)ε and m positive integer (see e.g Montgomery and Vaughan(2006)[Corollary 10.8], see also Cohen (2007)[Corollary 10.2.15 and Deﬁnition 10.2.16]), as well as s = 0 in case χ is a non principal (or non-trivial) even character (see Theorem 12.20 in Apostol (1976)). These zeros of L(χ, s) are the so-called trivial zeros. Furthermore from Cohen(2007)[Section 10.5.7], we get that L(χ, s) 6= 0 for

3 <(s) = 1. It follows that the nontrivial zeros of L(χ, s) are exactly those lying in the critical strip 0 < <(s) < 1. Now let us assume that χ is primitive (i.e. χ = χ0), then we have the following beautiful functional equation, discovered by Riemann in 1860 for the case k = 1 (Riemann zeta function) and worked out for general k by Hurwitz in 1882 (see e.g Montgomery and Vaughan(2006), Corollary 10.8):

s/2 (1−s)/2 k ΓR(s + η)L(s, χ) = ε(χ)k ΓR(1 − s + η)L(1 − s, χ). (10)

−s/2 η Here ΓR(s) := π Γ(s/2), η ∈ {0, 1} such that χ(−1) = (−1) , and ε(χ) is an explicitly computable complex number of modulus 1. It follows that there are inﬁnitely many zeros ρ with real part at least 1/2 (see Bombieri and Hejhal (1995)); in fact it seems that all zeros in the critical strip have real part equal to 1/2. Similar to the Riemann zeta function, there is a Generalized Riemann Hy- pothesis: Conjecture 1.2 (Generalized Riemann Hypothesis). For any Dirichlet character χ modulo k, the Dirichlet L-function L(χ, s) has all its non trivial zeros 1 on the critical line <(s) = . 2 Or, in other words, that L(s, χ), for a Dirichlet character χ modulo k, has 1 no zeros with real part diﬀerent from in the critical strip 0 < <(s) < 1, since 2 we can exclude non-trivial zeros outside.

2. Proof of Generalized Riemann Hypothesis

In this section, through Theorem 2.1 we prove that the Generalized Riemann Hypothesis is true. To this end, we will need to establish a series of Lemmata. The analytic-algebraic structure of Dirichlet L-functions was the key for the resolution of the Generalized Riemann Hypothesis. Let us ﬁrst record some immediate consequences from deﬁnition of Dirichlet character modulo k. For any integer n we have χ(n) = χ(n · 1) = χ(n)χ(1) by (iv), and since χ(n) 6= 0 for some n by (i), we conclude that χ(1) = 1. Next, if (n, k) = 1 then, using ((iv), (iii)) and Euler’s theorem which states that nϕ(k) ≡ 1 (mod k) with ϕ the Euler’s totient function (see e.g Theorem 5.17 in Apostol(1976)), we infer that

χ(n)ϕ(k) = χ(nϕ(k)) = χ(1) = 1, so that χ(n) is a ϕ(k)-th root of unity. Therefore, we get,

|χ(n)| = 1 if (n, k) = 1, (11) |χ(n)| = 0 if (n, k) > 1.

Now, we introduce some L-functions and some subsets of the complex plane. For any Dirichlet character χ, we introduce the Generalized prime Dirichlet

4 L-function P deﬁned by the series

X −s P (χ, s) = χ(p) p , s ∈ C (12) p∈P which is analytic for <(s) > 1. Indeed, the series converges absolutely when <(s) > 1. We recall that P is the set of prime numbers. For any Dirichlet character χ, we introduce P2 the L-function deﬁned by the series X 2 −s P2(χ, s) = χ(p) p , s ∈ C (13) p∈P which is analytic for <(s) > 1. For any Dirichlet character χ, we introduce also Q the L-function deﬁned by the series

∞ ! X X χ(p)r p−rs Q(χ, s) = χ(p)2p−2s , s ∈ r + 2 C p∈P r=0 1 which is analytic for <(s) > , indeed, the series converges absolutely when 2 1 <(s) > . 2 Let us denote by A the complex half plane

A = {s ∈ C : <(s) > 1}.

For any Dirichlet character χ, we denote also by Mχ the set

1 M = s ∈ \{1} : <(s) > ,L(χ, s) 6= 0 . χ C 2

Lemma 2.1. Let χ be a Dirichlet character. For all s ∈ A, we have

log L(χ, s) = P (χ, s) + Q(χ, s).

Proof. For any s ∈ A, thanks to (8) we get X log L(χ, s) = − log(1 − χ(p)p−s). (14) p∈P

Furthermore, for any p ∈ P we have for all s ∈ A.

∞ X χ(p)rp−rs − log(1 − χ(p)p−s) = r r=1 ∞ (15) X χ(p)rp−(r−2)s = χ(p)p−s + p−2s . r r=2

5 After plugging Equation (15) into (14), we get for all s ∈ A

∞ X X X χ(p)r p−rs log L(χ, s) = χ(p) p−s + p−2sχ(p)2 r + 2 p∈P p∈P r=0 = P (χ, s) + Q(χ, s), which concludes the proof

Now, we need to extend Theorem 1 established in Vassilev-Missana(2016) for the zeta function ζ and for integer s > 1 to Dirichlet L-functions and for s ∈ A. To this end, we need the following Lemma. Lemma 2.2. Let χ be a Dirichlet character. Let p 6= q two prime numbers, a ∈ N, b ∈ N, a ≥ 2, b ≥ 2 such that χ(p a) 6= 0, χ(q b) 6= 0 and s ∈ C with <(s) 6= 0. Then we have χ(p a) χ(q b) = (16) (p a)s (q b)s if and only if there exists k ∈ N∗ such that a = kq and b = kp. Proof. Let us assume that (16) holds. We take the module in Equation (16) to obtain |χ(p a)| |χ(q b)| = . |(p a)s| |(q b)s|

Thanks to (11) and since |(p a)s| = (p a)<(s), |(q b)s| = (q b)<(s), we deduce that 1 1 = . (p a)<(s) (q b)<(s)

Then, we infer that

q b <(s) = 1. p a which implies since <(s) 6= 0

q b = 1. p a Then we get qb = pa. (17)

Since p 6= q, from (17) we deduce that q divides a and hence there exists k ∈ N∗ such that a = kq. By plugging the new value of a in (17), we deduce b = kp. Then, we conclude the ﬁrst part of proof.

6 Let us assume now that there exists k ∈ N∗ such that a = kq and b = kp. On one hand, we have

χ(p a) χ(p k q) = . (p a)s (p k q)s

On the other hand, we have

χ(q b) χ(q k p) = . (q b)s (q k p)s

Then we infer that χ(p a) χ(q b) = , (p a)s (q b)s which completes the proof

Owing to Lemma 2.2, Lemma 2.3 appears as an extension of Theorem 1 of Vassilev-Missana(2016). Although the proof of Lemma 2.3 is similar as the one given in Vassilev-Missana(2016), we give here the details of the proof as it is at the heart of the Theorem obtained in this paper. For this, we borrow the arguments used in Vassilev-Missana(2016). Lemma 2.3. Let χ be a Dirichlet character. For s ∈ A, we have

2 (1 − P (χ, s)) L(χ, s) − (P2(χ, 2s) − 1)L(χ, s) = 2.

Proof. Let P be the set of all composite numbers (the numbers which are not prime) strictly greater than one. We introduce P the L-function deﬁned by

X χ(m) P (χ, s) = , s ∈ , (18) ms C m∈P which is analytic for <(s) > 1. We observe that we can re-write P (χ, s) as follows X χ(m) P (χ, s) = . ms m∈P,χ(m)6=0

From (7), (12) and (18), we have for all s ∈ A

P (χ, s) + P (χ, s) = L(χ, s) − 1. (19)

7 For any s ∈ A, we consider P (χ, s)(L(χ, s) − 1) and then we get

  ∞ ! X χ(p) X χ(n) P (χ, s)(L(χ, s) − 1) =  ps  ns p∈P n=2    ∞  X χ(p) X χ(n) =  ps   ns  p∈P,χ(p)6=0 n=2,χ(n)6=0 (20) X χ(p) χ(n) = ps ns p∈P,χ(p)6=0,n∈N,n≥2,χ(n)6=0 X χ(p n) = , (p n)s p∈P,n∈N,n≥2,χ(p n)6=0 where we have used (iv). Furthermore, since 2 and 3 are prime numbers, we observe that for any composite number m > 1 there exists p ∈ P and n ∈ N, n ≥ 2 such that m = pn and χ(m) 6= 0 if and only if χ(p) 6= 0 and χ(n) 6= 0 thanks to (iv). X χ(p n) Then for any s ∈ A, the sum yields P (χ, s) but also, (p n)s p∈P,n∈N,n≥2,χ(p n)6=0 some repeating terms will be there. Thanks to Lemma 2.2, for s ∈ A we deduce that the sum of these repeating terms is given by

def X χ(kpq) S(χ, s) = . (21) (kpq)s ∗ 2 k∈N ,(p,q)∈P ,χ(kpq)6=0,p

X χ(kpq) S(χ, s) = (kpq)s ∗ 2 k∈N ,(p,q)∈P ,p

8 where def X χ(p)χ(q)) J (s) = . (24) χ ps qs (p,q)∈P2,p

(s) P (χ, s)(L(χ, s) − 1) = P (χ, s) + L(χ, s) Jχ , which is re-written as

(s) P (χ, s) = P (χ, s)(L(χ, s) − 1) − L(χ, s) Jχ . (25)

By plugging (25) into (19), we obtain that for all s ∈ A

(s) P (χ, s)L(χ, s) − L(χ, s) Jχ = L(χ, s) − 1. (26)

(s) It remains only to ﬁnd Jχ , but we have for all s ∈ A:

 2 X χ(p) (P (χ, s))2 =  ps  p∈P X χ(p)2 X χ(p)χ(q) = + 2 p2s ps qs p∈P (p,q)∈P2,p

Then, we get that for all s ∈ A

(P (χ, s))2 − P (χ, 2s) J (s) = 2 . (27) χ 2 After replacing (27) into (26) we obtain that for all s ∈ A

2 (1 − P (χ, s)) L(χ, s) − (P2(χ, 2s) − 1)L(χ, s) = 2, which concludes the proof. In the Lemma below, by means of analytic continuation, we extend the results obtained in Lemmata 2.1 and 2.3 on the complex half plane A to Mχ. Lemma 2.4. Let χ be a Dirichlet character. We get that P (χ, ·) is analytic on Mχ. Furthermore for all s ∈ Mχ, we have 1 1 = (1 − P (χ, s))2L(χ, s) − (P (χ, 2s) − 1)L(χ, s) . 2 2 and

log L(χ, s) = P (χ, s) + Q(χ, s).

9 Proof. Thanks to Lemma 2.3, we get that for any s ∈ A 1 1 = (1 − P (χ, s))2L(χ, s) − (P (χ, 2s) − 1)L(χ, s) . (28) 2 2 Thanks to Lemma 2.1, we have for any s ∈ A

log L(χ, s) = P (χ, s) + Q(χ, s). (29)

def 1 Since Q(χ, ·) is analytic on the complex half plane B = s ∈ C; <(s) > 2 ⊃ Mχ and log L(χ, ·) is analytic on Mχ, by means of analytic continuation, we infer from Equation (29) that P (χ, ·) is analytic on Mχ and since s 7→ P2(χ, 2s) is analytic on B then Equations (28) and (29) are valid on Mχ. Then we conclude the proof. In the following Lemma, we derive a new equation satisﬁed by any Dirichlet L-function. This equation is the key point in obtaining the proof of our Theorem 2.1.

Lemma 2.5. Let χ be a Dirichlet character. For all s ∈ Mχ, we have 1 1 = L(χ, s)(log L(χ, s))2 − (1 + Q(χ, s))L(χ, s) log L(χ, s) 2 1 1 + (1 + Q(χ, s))2 − (P (χ, 2s) − 1) L(χ, s). 2 2 2

Proof. Thanks to Lemma 2.4, we have for all s ∈ Mχ, 1 1 1 = (1 − P (χ, s))2L(χ, s) − (P (χ, 2s) − 1)L(χ, s). (30) 2 2 2 and P (χ, s) = log L(χ, s) − Q(χ, s). (31)

After plugging Equation (31) into (30), we obtain for all s ∈ Mχ, 1 1 1 = (1 + Q(χ, s) − log L(χ, s))2L(χ, s) − (P (χ, 2s) − 1)L(χ, s) 2 2 2 which yields 1 1 = L(χ, s)(log L(χ, s))2 − (1 + Q(χ, s))L(χ, s) log L(χ, s) 2 1 1 + (1 + Q(χ, s))2 − (P (χ, 2s) − 1) L(χ, s). 2 2 2

Then, we conclude the proof.

Now, we turn to the proof of our Theorem.

10 Theorem 2.1. For any Dirichlet character χ modulo k, the Dirichlet L-function 1 L(χ, s) has all its non trivial zeros on the critical line <(s) = . 2 Proof. Let χ be a Dirichlet character. From Cohen(2007)[Section 10.2.4], we have that all the non trivial zeros of L(χ, ·) lie in the critical strip :

S = {s ∈ C : 0 < <(s) < 1}. (32) From the functional equation (10) and the elementary property L(χ, s) = L(χ, s) we get that the zeros of L(χ, s) in S are symmetric with respect to the critical 1 line <(s) = , then to prove our Theorem it suﬃces to show that there is no 2 zeros of L(χ, ·) in the following critical strip : 1 U = {s ∈ : < <(s) < 1}. (33) C 2

Then for a contradiction, let us assume that there exists s0 ∈ U such that L(χ, s0) = 0. Due to the analyticity of the Dirichlet L-function L(χ, ·) on U, we infer that the zeros of the Dirichlet L-function L(χ, ·) are isolated and then there exists Us0 ⊂ U an open neighbourhood of s0 such that Us0 \{s0} contains no zeros of the Dirichlet L-function L(χ, ·).

That means that for all s ∈ Us0 \{s0}, |L(χ, s)| > 0. (34)

We thus observe that Us0 \{s0} ⊂ Mχ, then thanks to Lemma 2.5 for all s ∈

Us0 \{s0}, 1 1 = L(χ, s)(log L(χ, s))2 − (1 + Q(χ, s))L(χ, s) log L(χ, s) 2 1 1 (35) + (1 + Q(χ, s))2 − (P (χ, 2s) − 1) L(χ, s). 2 2 2

Since the complex functions s 7→ Q(χ, s), s 7→ P2(χ, 2s) are analytic on 1 s ∈ ; <(s) > ⊃ U and the Dirichlet L-function L(χ, ·) is analytic on C 2 s0

U ⊃ Us0 then we obtain 1 2 1 lim (1 + Q(χ, s)) − (P2(χ, 2s) − 1) L(χ, s) s→s0,s∈Us0 \{s0} 2 2 1 1 = (1 + Q(χ, s ))2 − (P (χ, 2s ) − 1) L(χ, s ) 2 0 2 2 0 0 = 0. (36) where we have used the fact that L(χ, s0) = 0. Furthermore, for any α > 0 we have lim x| log x|α = 0 and since s 7→ x→0,x>0

|L(χ, s)| is continuous on Us0 we get also

lim |L(χ, s)| = |L(χ, s0)| = 0. (37) s→s0,s∈Us0 \{s0}

11 We thus deduce that for any α > 0

lim |L(χ, s)| | log |L(χ, s)||α = 0, (38) s→s0,s∈Us0 \{s0} thanks also to (42). By using the deﬁnition of the complex logarithm function, we have for all s ∈ Us0 \{s0} log L(χ, s) = log |L(χ, s)| + iarg(L(χ, s)), (39)

where arg(L(χ, s)) ∈] − π, π]. We thus obtain that for all s ∈ Us0 \{s0} |L(χ, s)(log L(χ, s))2| = | log L(χ, s)|2|L(χ, s)| = ((log |L(χ, s)|)2 + (arg(L(χ, s)))2)|L(χ, s)| (40) ≤ (| log |L(χ, s)||2 + π2)|L(χ, s)|

Then thanks to (37) and (38) used with α = 2, from (40) we infer that

lim |(log L(χ, s))2L(χ, s)| = 0. (41) s→s0,s∈Us0 \{s0}

We have also that for all s ∈ Us0 \{s0} |L(χ, s) log L(χ, s)| = | log L(χ, s)| |L(χ, s)| 2 2 1 = ((log |L(χ, s)|) + (arg(L(χ, s))) ) 2 |L(χ, s)| (42) ≤ (| log |L(χ, s)|| + |(arg(L(χ, s))|)|L(χ, s)| ≤ (| log |L(χ, s)|| + π)|L(χ, s)|

Then thanks again to (37) and (38) used with α = 1, from (42) we infer that

lim |L(χ, s) log L(χ, s)| = 0. (43) s→s0,s∈Us0 \{s0} Owing to (36), (41) and (43), after taking the limit in Equation (35) as s → s0, s ∈ Us0 \{s0}, we obtain that 1 = 0 which leads to a contradiction. Hence, we deduce that there is no zeros of the Dirichlet L-function L(χ, ·) in U and then we conclude the proof.

3. Conclusion In this paper, we have proved the Generalized Riemann Hypothesis and as a immediate consequence by taking the Dirichlet character χ = 1 we get also the proof of Riemann Hypothesis. Thus, our proof yields to the best possible bound for the error of the Prime Number Theorem (see (6)). It leads to the veracity of several theorems whose statements begin by assuming the Rie- mann Hypothesis or the Generalized Riemann Hypothesis is true (see Borwien et al.(2008)). It leads also to the veracity of theorems which are equivalent to Riemann Hypothesis or Generalized Riemann Hypothesis (see Borwien et al. (2008)). In particular, thanks to our result, the Goldbach’s Weak Conjecture holds (see Deshouillers et al.(1997)).

12 References P. Borwien, S. Choi, B. Rooney, and A. Weirathmueller. The Riemann Hypoth- esis: A Resource for the Afficionado and Virtuoso Alike. Springer, 2008. K. Sabbagh. The Riemann Hypothesis: The Greatest Unsolved Problem in Math- ematics. Farrar and Straus and Giroux, 2002. M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover, 1964. E. M. Stein and R. Shakarchi. Complex Analysis, Princeton Lectures in Analysis II. Princeton University Press, Princeton and Oxford, 2003. J.B Conrey. The Riemann Hypothesis. Notices of the American Mathematical Society, 50:341–353, 2003. B. Riemann. Uber¨ die Anzahl der Prinzahlen unter einer gegebener gröse. Monastsber. Akad. Berlin, pages 671–680, 1859. H. M. H.M. Edwards. Riemann’s zeta function. Acad. Press. New York, 1974. E. C. Titchmarsh. The Theory of the Riemann Zeta-function, 2nd edition (revised by D. R. Heath-Brown). Oxford University Press, Oxford, 1986. H. L. Montgomery and R. C. Vaughan. Multiplicative Number Theory I: Clas- sical Theory. Cambridge University Press, 2006. H. von. Koch. Sur la distribution des nombres premiers. Acta Math., 24:159–182, 1901. H. Cohen. Number theory. Vol. II. Analytic and modern tools, volume 40. Grad- uate Texts in Mathematics. Springer New York, 2007. T. M. Apostol. Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics. Springer-Verlag, 1976. H. Davenport. Multiplicative Number Theory, 2nd edition, revised by H. Mont- gomery, Graduate Texts in Mathematics, volume 74. Springer-Verlag, New York, 1980. W. J. Ellison and F. F. Ellison. Prime Numbers. John Wiley & Sons, New York, 1985. J. P. Serre. A Course in Arithmetic. Springer-Verlag, New York, 1986. E. Bombieri and D. A. Hejhal. On the distribution of zeros of linear combinations of euler products. Duke Math. J., 80(3):821–862, 1995. M. Vassilev-Missana. A note on prime zeta function and Riemann zeta function. Notes on Number Theory and Discrete Mathematics, 22(4):12–15, 2016. J. M. Deshouillers, G. Effinger, H. J. J. te Riele, and D. Zinoviev. A complete Vinogradov 3-primes theorem under the Riemann Hypothesis. Electronic Re- search Announcements of the American Mathematical Society, 3:99–104, 1997.