The Riemann Zeta Function and Its Analytic Continuation

The objective of this dissertation is to study the Riemann zeta function in particular it will examine its analytic continuation, functional equation and applications. We will begin with some historical background, then define of the zeta function and some important tools which lead to the functional equation. We will present four different proofs of the functional equation. In addition, the ζ(s) has generalizations, and one of these the Dirichlet L-function will be presented. Finally, the zeros of ζ(s) will be studied.


Introduction
One of the oldest branches of mathematics is number theory. Many mathematicians have acknowledge the significance of complex analysis and applied it to number theory. The relationship between *Corresponding author: E-mail: ahlam.alhadbani@gmail.com; number theory and complex analysis is called Analytic Number Theory. The story begins formally with Euler's theorem,which was proved (1.1) to be divergent in 1737. Following this Riemann considered the Euler's definition from the perspective of its complex argument.
In 1859, Riemann extended the definition of Euler's zeta function from real variables to complex variables except a simple pole at s = 1 with residue 1. In his paper "On The Number of Primes Less Than a Given Magnitude". Riemann forced the zeta function to be a meromorphic function, and he proved the functional equation by using the analytical tools.
The Riemann zeta function is one of the most essential functions in mathematics. Its applications include many areas of study such as number theory and other sciences. This function was not created by Riemann but it is named after him since he developed the zeta function and proved the functional equation as well as demonstrating the significant relationship between the distribution of prime numbers and the Riemann zeta function.
Some of results required presentation because they will be used frequently throughout this dissertation.
The aim of this dissertation is to study the Riemann zeta function, in particular its analytic continuation ,functional equation and applications.
This dissertation is divided into seven sections. The first section, briefly gives the historical background with particular attention paid to the development of the zeta function. In the second section, some definitions and theorems related to the analytic continuation are provided along with some elementary asymptotic formula with out proofs. In the third section, provides several important analytic tools which help to prove the functional equation and play an important role in reaching the dissertation's goal. In the fourth section, the definition of the zeta function is stated in two ways, as the Dirichlet series and as the Euler product. In the fifth section, the functional equation is proved using four different methods. Then, In the sixth section, the zeta function has generalizations and one of these the Dirichlet L-function is presented with its analytic continuation. In addition, the function equation for the Dirichlet L-function is provided.Finally, studies the zeros of the zeta function and the Dirichlet L-function.

Background
In the 14th century ,the harmonic series ,ζ(1) was proved to be divergent. Then,in the first half of the 18th century, Euler was the first mathematician to introduce the zeta function. Euler's zeta function is defined for any real number Euler only defined ζ(s) on R. It means that Euler gave birth to ζ(s). Then, in the 19th century , Riemann improved the zeta function by using complex analysis. He extended the Euler definition from real variables to complex variables. Additionally, he proved the analytic continuation of the zeta function, hence obtaining the functional equation. Riemann considered the zeta function to be a complex function. He published his paper which is one of the most effective studies of modern mathematics in 1859. Moreover,by using complex analysis, Riemann connected the zeros of ζ(s) and the distribution of the prime numbers. However, before Riemann conjectures were proved or a results about primes extrapolated, he proved the two main consequences: (1) The zeta function can be meromorphic continued over the whole s-plane with a simple poles at s = 1 ,such that ζ(s) − 1 s−1 is an integral function. (2) The zeta function satisfies the reflection, The left hand side of this function is an even function on s − 1 2 , can be conclude the proprieties of the zeta function for σ > 0 from the proprieties for σ > 1 by the functional equation. Particularly, when σ < 0 the zeros of the zeta function are poles of Γ s 2 s at negative even integers, these are called the trivial zeros. The remain in a plane, when 0 < σ < 1 is called the critical strip. Riemann made several other remarkable conjectures (i) the zeta function has all its zeros in the critical strip (ii) ξ(s) is an integral function defined as which has no pole when σ ≤ 1/2 and this function is an even function of s − 1 2 , also it has the product ξ(s) = e A+Bs λ 1 − s λ e s/λ such that A and B are absolutely constants and λ runs through the zeros of the zeta function in 0 ≤ σ ≤ 1. Note that the most information is taken from Awan [1],Davenport [2] and Moros [3].

Analytic Number Theory
This section recall one of the most significant part of complex analysis that will be help to know the basic of the analytic number theory. Most of results will be taken from Brown et al. [4], Lang [5] and Titchmarsh [6].

Uniform convergence
Definition 3.1. If the partial sum Sn(x) = ∞ n=0 fn(x) exists, the series ∞ n=0 fn(x) is uniformly convergent in (a,b), if every ≥ 0 such that is small. There exist a number n1 depending on however,n1 not depending on x .

Identity theorem
If a region R has two analytic and these functions have the same values for all points within R , both functions are identical everywhere within R . This result can be used to extend functions from the real axis to the complex plane.

Analytic functions
(ii) f is analytic on B, it has derivative at all points in B. (iii) If the function f is analytic in a neighborhood of b0 then f is analytic of b0. (iv) The entire is a function on the whole complex plane.

Remark:
If the function f has derivative, then f is continuous. However, if f is a continuous function, it does not have to be differentiable.

Analytic continuation
Definition 3.3. Suppose D1 and D2 are domains in the complex plane; the intersection of these domains ,D1 ∩ D2, is the set of all common points between D1 and D2. The union of these domains ,D1 ∪ D2, is the set of all points in D1 and D2. Now, let D1 ∩ D2 = φ ; this is also connected. The functions f1, f2 are analytic over the domains D1, D2, respectively, such that f1 = f2 on D1 ∩ D2. Hence, we can call f2 an analytic continuation of f1 in the second domain.
According to Brown et al. [4] the essential theorem from the theory of complex variables is: A function that is analytic in a domain D is uniquely determined over D by its values in a domain, or along a line segment, contained in D.
The definition of F(b) analytic over D1 ∪ D2 Since, f1 = f2 over D1 ∩ D2, F is given by f1 and f2 over the domains D1 and D2 respectively. Based on the theorem this is a holomorphic function.
Since F is analytic in the union of D1 and D2 the function is uniquely determined by f1 on the domain D1 furthermore, it is uniquely determined by f2 on a domain D2.
Thus, F (b) is analytic continuation over the union of D1 and D2 of either f1 or f2.

Some elementary asymptotic formula
Definition 3.4. (The "big oh" notation) Moreover, it satisfies this equation: This means that f (y) = O(h(y)) for l ≥ a implies The next theorem provides a number of asymptotic formula. In part (a) C is Euler's constant and it defined as In part (b) the Riemann zeta function ζ(s) defined by And by Proof: See [7].
Note that these results are taken from Apostol [7] 4 Some Analytic Tools As mentioned previously, Euler defined ζ(s) only for real variables. However, Riemaan extended this definition to the domain of complex variables; to achieve this, Riemann studied the analytic continuation of ζ(s). The functional equations are extremely important in analytic continuation.
To prove the functional equation requires certain some analytical tools and these will be provided and some of their properties discussed. This in turn will lead to several proofs of the functional equations. This section divides into six subsections, we will begin with the gamma function ,the theta function and the mellin transform. Then, the Dirichlet series and Abel summation (Partial summation) will be presented. Finally,the Fourier transformation will be laid out.

The gamma function
The gamma function is an important function for analytic number theory. It is useful function when dealing with the theory surrounding the zeta function. In particular, it is an extension of the factorial function to Re(s) > 0. It is also convergent.
Let we introduce this function : The first to denote this function as Γ(s) was Legendre but the first to introduce it was Euler. However, Euler defined the function somewhat differently.
For σ > 0 the term (−e −t t s ) ∞ 0 tends to zero. However, extending Γ(s) to the entire complex plane needs the analytic continuation. The gamma function has several convenient properties support this.
where the oh big depend on Thus, for > 0 this integral shows a continuous function.
The last propositions are taken from Chandrasekharan [10].

The theta function
Also, we can be written as The relationship between θ(t) and w(t) as 2w(t) = θ(t) − 1 The theta function θ(t) satisfies the function equation: Before proving this propitiation it is necessary to we introduce: The Poisson Summation Formula Hence, We have It can be shown that and hence It has already been shown that It follows that suppose θ(t) = ∞ n=1 e −πn 2 x then,

Mellin transform
The Mellin Transform is an integral transform defined as Examples: (1) As we know the gamma function is defined as Directly, we show that It is obvious that this is analytically continues Γ(s) to the right half plane Re(s) > 0.
By mellin transform we have that Hence, from the definition of the zeta function and the gamma function we find that Remark: Relating the theta function and the zeta function is required, particularly as there is a relationship between the Mellin function and the zeta function of the theta function. This relationship will appear in the third method of deriving the functional equation. Note that the Mellin function information is taken from Oosthuizen [13].

Abel summation (partial summation)
Theorem 4.11. Suppose φ(x) is a function whose values belong to C and this function has continuous derivative on the interval [a,b]; let cm be arbitrary in C and C(x) = a≤m≤x cm Then, also, we can be written as which yields the required result, after canceling.
The partial summtion results are taken from Karatsuba and Voronin [9].

Fourier transform functions
If the function f (x) is an even, then the Fourier integral represent as this formula called the Fourier's cosine.
Similarly, If the function f (x) is an odd, then we obtain that this formula called the Fourier's sine. If we say Then, (4.4) provides: Hence, the functions f (x) and h(x) have a reciprocal relationship. Likewise, from 4.5 we find that the reciprocal formula and cos xt g(t) dt

Integration of fourier integral
It holds over the interval (0, ∞) for any integral function sin ξy cos yt y dy For every y and t, the uniform convergent and inner product on the right is bounded. Therefore, it holds that Now, we write the sign of ξ − t and This formula is a bounded function of U. Therefore, by using Lebesgue's convergence theorem as Y goes to infinity ∞ 0 sin ξy cos yt y dy = π 2 Note that These results were obtained from Titchmarsh [6].

Definition of ζ(s)
The Riemann zeta function can be defined by a simple formula in two representations, the first representation of ζ(s) by the Dirichlet series Hence, it uniformly converges. Moreover, the Riemann zeta function is analytic for σ > 1.

The euler product
where, this product is taken over every prime number.
Proof. Suppose X ≥ 2, the function ζX (s) is defined as with all factors on the right hand side, allowing the term to be presented by a geometric series formula 1 p s such that every geometric series is convergent. This allows term by term multiplication and substitute the right hand side with this equation: such that p1 < p2 < ....... < pj and pj represents all the prime numbers up to X.
We use a fundamental theorem of arithmetic (unique factorization of integers) and we see that every positive integer number n ≤ X can be represented as when the 1...... j are positive integers. As a result, can be taken the right side of (5.3) as where the ' stands for summation over those natural numbers (n greater than X) whose primes divisors are all less than or equal to X. The upper bound of this sum is The final formula provides an upper bound for the sum. Combining the definition of the zeta function ζX (s) with (5.2),(5.1)and (5.4) gives If X goes to +∞ we obtain X 1−σ goes to +∞ such that σ > 1. Thus, we proved that Now, we will present an important properties of Riemann zeta function in terms of analytic continuation.
Initially, we will prove fn(s) to be convergent, then we will show that For σ > 0 fn(s) and 1 s−1 are analytic , then ∞ n=1 fn(s) + 1 s−1 is an analytic continuation of the zeta function.
such that ξ(s) is an analytic continuation on the entire plane and that ξ(s) satisfies Proof. Using the definition of the zeta function and the gamma function. Since and Applying (5.6) and (5.7) in (5.5) gives Dividing the integral We use the corollary (4.1.5) We obvious that K(s) is an entire function. In addition, 1 s and 1 s − 1 are analytic when s = 0, 1, is an analytic continuation of ξ(s).

Functional Equations and Analytic Continuation
One of the aims in this dissertation is to prove the functional equation of ζ(s). In this section, four different proofs of the functional equation are provided. Note that these proofs are taken from Davenport [2],Titchmarsh [14] and Chandrasekharan [10]. More methods are available in [14].

First method
Proof. We will prove this theorem depending on the summation formula.
Assume φ(x) to be a function which is continuously differetiable on [a, b] a; [x] denotes the greatest integer less than x.
Using Euler summation formula gives With respect to (a, b]; this formula is clearly additive. It is sufficient to assume that n ≤ a < b ≤ n + 1.
Take σ > 1 , where a = 1 , and b goes to ∞ and add 1 to both sides hence we obtain: Titchmarsh noted that the numerator x−[x]+ 1 2 is bounded and, when σ > 0, the integral in (6.4) is convergent. This integral is uniformly convergent in any finite region to the right of σ = 0. Hence, for σ > 0 , it defines the analytic function of s regularly. The right hand side gives the analytic function ζ(s) up to σ = 0 and it is clear that a simple pole at s=1 with residue 1.
If σ > −1 , the integral (6.4) is convergent. Therefore, x s+1 dx f or − 1 < σ < 0 (6.8) [x] − x + 1 2 has a Fourier series expansion If x is not integer, we will substitute the series (6.9) in equation (6.8) to gives This permitted for −1 < σ < 0, but, for σ < 0 the right hand side is analytic for all value of s. Therefore, this gives the analytic continuation (not just for −1 < σ < 0), which means it provides the analytic continuation throughout the plane. This is sufficient to prove to justify the term-wise integration.
The series in (6.9) is bounded convergent, so the term-wise integration is absolutely justified over any finite range. Then where it is exactly proved from (6.11) and (6.12) * Hardy applied a similar argument in the first method , However not to it self, to this function Hardy's proof: one can see the proof in Titchmarsh [14].

Second method
The fundamental formula is: Proof. to prove this formula, start with Γ(s) = ∞ 0 x s−1 e −x dx (6.14) By replacing the variable By the definition of ζ(s) we find that Now, we assume the complex integral such that the contour C containing of the positive real axis from ∞ to ρ for 0 < ρ < 2π , the circle |z| = ρ and the positive real axis from ρ to ∞. On the circle , Therefore, for σ > 1 the integral round this circle goes to 0 as ρ → 0 Hence on letting ρ → 0 , we obtain If s ∈ N and since z e z − 1 where (B1, B2...Bernoulli numbers).

Let the integral
We take the integral along the contour Cn as in the diagram, There is an integral between C and Cn which has poles at the points ±2iπ, ....., ±2inπ the residue at −2miπ is (2mπe Now, consider σ > 0 as n goes to infinity. On the contours Cn, the function 1 (e z − 1) is bounded.
Therefore, the integral round Cn goes to zero and we get Note that the diagram taken from Chandrasekharan [10].
* When Re(s) < 0 and therefore by analytic continuation for ∀s By taking Γ s 2 Γ s + 1 2 = √ π 2 s−1 Γ(s) and changing the variables from s to 1 − s we obtain that

Third method
Proof. This method proves the equation (6.17) by the gamma function We add sum from both sides Since he ζ(s) defined as We apply (6.20)in(6.19) It follows that Since we have For any s ,this integral is convergent and this formula provides the analytic continuation. If we replace s by s − 1 the right side is unchanged. Hence, we obtain

Fourth method
Proof. This method depends on self-reciprocal function. If σ > 1 could be written as For σ greater than zero and analytic continuation (6.13) holds. Moreover, if 0 < σ < 1 we have Since the function f (x) = 1 is self-reciprocal for the sine transforms by (4.6) f (x) = 2 π ∞ 0 f (y) sin xy dy Let x = ξ (2π) and ξ = x √ 2π in (6.24) gives as y tends to 0 and equal O(y −1 ) as y tends to ∞, we get that The same way, when goes to 0.

The dirichlet L-function
There are many ways in which the Riemann zeta function has been generalized and this dissertation will introduce only one of these generalizations. This section will give the definition of the Dirichlet L-function, an analytic continuation of L(s, χ) and the functional equation of L(s, χ).
Recall the basic definition: The function L(s, χ) was introduced by Dirichlet in 1837 , it defined as This function is a holomorphic function for the real part of complex variable s greater than 1.
Moreover, the L-function can be written as an Euler's product for ζ(s) The proof of (7.1) is precisely the same proof of theorem (5.1) Assume χ(n) = χ1(n) where χ1(n) is the principal character modulo and if For the real part Re(s) > 1, the L(s, χ) connected with a character χ modulo .
By this relation We obtain that The only difference between L(s, χ1) and ζ(s) by a simple factor.
7.2 Analytic continuation of L(s, χ 1 ) for Re(s) > 0 By following next lemma , the analytic continuation is obtained. such that χ(n) is non principal character modulo . Then, for Re(s) greater than zero we have that Proof. Suppose Re(s) > 1 and N ≥ 1 We use the partial summation and we get Such that c(x) = S(x) − 1 and x ≥ |c(x)|, let N goes to infinity we get that The lemma is proved. Suppose χ be any Dirichlet character modulo and L(s, χ) be connected L-function then: (i) If a Dirichlet characters χ = χ1, such that χ1 is the principal character modulo therefore, L(s, χ) is analytic in Re(s) > 0.
(ii) If a Dirichlet characters χ = χ1,therefore, L(s, χ) has a simple pole at s = 1 and is analytic at all remain points in Re(s) > 0.
As a result, for Re(s) greater than 1 we have By the definition of L(χ, s) and the equation (7.7) we find Dividing the integral into two parts Therefore, (7.13) provides an analytic continuation of L(s, χ) onto the whole s-plane, L(s, χ) is regular;moreover,if s is changed by1 − s and χ by χ, the right hand side of (7.13) is multiplied by .

The Zeros of ζ(s)
According the theory of the Riemann zeta function, the critical strip is the region 0 ≤ σ ≤ 1 of the complex plane. The critical line is the line σ = 1/2. The trivial zeros of the Riemann zeta function are the real zeros of ζ(s) at −2, −4, −6, ..., −2n.. these are negative even numbers. The non-trivial zeros are the complex of ζ(s). This section will give the theorems for trivial and nontrivial zeros. However, before this it will provide a few important basic theorems which help to reach the desired results. These results were obtained from Gleinig and Bars [16] and Stein and Shakarchi [17]. Chapter 7 in [17] also has more details. The function Γ(s) initially defined for Re(s) > 0 has an analytic continuation to a meromorphic function on C whose only singularities are simple poles at the negative integers. The Riemann zeta function only has zeros at s = −2, −4, −6, ... − 2n, .. and the complex numbers λn that lie on (0 < Re(s) < 1) the critical strip. Moreover, on the critical strip the zeros are situated symmetrically respecting to the real axis and to the point 1/2.
Proof. Define the Möbius function µ(s) as if n = p1....p and p1....p are distinct primes 0 Otherwise The relation between this function and Euler Product is From the inequality , we prove the zeta function has no zero when Re(s) greater than 1. In addition, for Re(s) less than zero and using the Riemann's function equation, gives ζ(s) with no zero apart from the zeros at s = −2, −4, −6, .... − 2n...
The next theorems,lemmas and proof in this section are taken from Stein and Shakarchi [17] and Riffer-Renert [18].
Theorem 8.5. The ζ(s) has no zero when Re(s) equals 1.
The proof of this theorem need some work with additional theorems and requires the statement of several important lemmas. An essential theorem about zeros and poles is also given.
Lemma 8.6. If the real part Re(s) greater than 1, then Cn n s such that Cn ≥ 0. Proof. Now, we will prove theorem (8.5)by contradiction.
let we have a point s0 = (1 + it) such that ζ(1 + it) = 0 for t = 0, we know that the zeta function is holomorphic at the point (1 + it), it must disappear at least to order 1 at (1 + it), by theorem (8.9) there exist a constant C, it is greater than 0 Therefore, |ζ(σ + it)| 4 C(σ − 1) 4 ≤ 1 as σ goes to 1 Likewise, by theorem (8.10) the zeta function has a simple pole at s = 1 and there exist a constant C , it is greater than 0 So |ζ(σ)| 3 C (σ − 1) 3 ≤ 1 as σ goes to 1 At the end, since the zeta function is holomorphic at σ + 2it , |ζσ + 2it| is remains bounded as σ tends to 1.
To understand and prove this lemma we need to review some theorems and corollary. We take these results from Karatsuba and Voronin [9]. On every compact subset of the s-plane ,this product is uniformly convergent , therefore it defines an entire function f (s).
such that A and B are belong to C (iii) If this series is convergent. Then ∃ c > 0 ,c is absolutely constant. one has log max |s|=R |G(s)| ≤ c(R + 1) Corollary 8.14.
Such that N belong to N and λ = 1 2 − [u].
such that δ greater than 0 and args between δ − π and π − δ.The constant O-big depends on δ.
By the corollary (8.14), when Re(s) ≥ Then, The order of this function at most 1, we obvious that it is enough to see that ξ(s) = 0 when σ > 1. This will follow if we show that ζ(s) = 0 when σ > 1.
By theorem (4.1) and when σ > 1 such that s = σ + it, we have that Taking the integral The proof of ξ(s) is the same as the proof of ξ(s, χ). We use (8.1) it easy to show that ξ(0) does not equal 0 and ξ(1) also does not equal 0. Since, Γ(s/2) = 0. Therefore, all of zeros of the product (8.3) are zeros of the Riemann zeta function ζ(s), for ξ(s, χ). Note that we take the proof from Karatsuba and Voronin [9] and Kumchev [19].

Theorems relating to the zeros of ζ(s)
By analytic continuation of ζ(s),the ζ(s) can represent as For σ > 0 and N equal 1 we find that Multiplying both sides by (s − 1) for (8.4) and (8.7), then take limit when s tends to 1, we get Then If |γn − t| > 1 hence, we obtain that

Results of the functional equation for L(s, χ)
The basic consequences focusing on the zeros of L-function are similar to the zeros of the Riemann zeta function.
If χ is an even character and theorem (7.3) we have that When Re(s) greater than 0, therefore that Re(1 − s) greater than or equal 1.We find that the only zeros of L(s, ξ) are poles of Γ(s/2) at negative even integers.