This paper provides an introduction to the theory of complex analysis and applies it to prove various fundamental properties of the Riemann zeta function. Specifically, integral identities for the zeta function are derived, its zeros are analysed both on and off the critical strip, and with the aid of a python script, some nontrivial zeros are approximated. The first section of the paper discusses the necessary background knowledge required to understand the findings in the paper. It covers holomorphic functions and Cauchyâ€™s theorem regarding contour integrals. It also provides some examples for each topic. Section 3 discusses the zeta function and the gamma function, and it states a relationship between the zeta and gamma functions. This section establishes a base to study the zeros of the zeta function. Section 4 is on the zeros of the zeta function, and provides proof that the zeta function has no zeros in some regions, some trivial zeros, and the non-trivial zeros on the critical strip. Section 5 explains how the author wrote a program to compute the zeros of the zeta function on the critical line. The data collected in this paper will add to the list of the roots of the Reimann zeta function, and the program provides a new way of empirically computing the zeros. The plot of the function may provide additional information about the function, and questions such as â€˜why is the shape of the plot a cardioid?â€™ can be asked to investigate further the zeta function.

## Introduction

The zeta function was first studied by Leonhard Euler. He was interested in its properties as a function that could be used to calculate the sums of infinite series. Euler discovered that the zeta function could be used to calculate the sum of the series 1 + 2 + 3 + 4 + …, which is equal to -1/12. He also showed that the zeta function has a simple pole at 1 with residue^{1}. Eulerâ€™s work on the zeta function lay dormant for many years until it was rediscovered by Bernhard Riemann in the 19* ^{th }*century. Riemann was interested in the zeta function as a way to understand the distribution of prime numbers. He showed that the zeta function has an infinite number of zeros, all of which lie on the line 1/2 + it. He also showed that the zeta function is connected to the prime number theorem, which states that the density of prime numbers is .

The zeta function has since been studied by many mathematicians and has found applications in a variety of fields, including physics, finance, and cryptography. The zeta function has many interesting properties, and the zeros of the zeta function are especially interesting. The zeta function has a close connection to the prime numbers and the zeros of the zeta function can give us information about the distribution of the prime numbers.

Eulerâ€™s work indicated a possible connection of the zeta function to the prime numbers. Riemann then discovered that the roots, of the zeta function, are what connect the function to the prime numbers. He hypothesized that the only zeros of the zeta function inside the critical strip would lie on the critical line, or the line where .

This paper portrays the authorâ€™s elementary understanding of complex analysis and that of the zeta function. The paper presents a list of the zeros of the zeta function that the author generated by writing a program. Employing the zeta function, we write a program that generates an array of the list of zeros of the zeta function. A final evaluation of the list revealed that the zeros found were accurate when compared to what other mathematicians have found.

The first section of the paper is the Introduction, which displays the paperâ€™s findings and results. The second section very briefly goes over some concepts in complex analysis. It covers some of the prerequisites needed to read and understand the paper. Section 2.1 discusses holomorphic functions. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of every point in its domain. An alternate procedure is to use the Cauchy-Reimann Equations, which are not mentioned in the paper but are explained in full detail in the book Complex Analysis^{2}

The section then includes some examples of functions that can and cannot be considered holomorphic. Additional examples are given in^{2}. The section then discusses Cauchyâ€™s Theorem and contour integration. Section 3 provides a proof that if then

This equation establishes a relation between the zeta function and the gamma function. The paper states that these two functions are very deeply interrelated and the study of the gamma function is crucial in the study of the zeta function and that of the prime numbers.

Section 4 discusses the zeros of the zeta function; it first proves that if then converges. The proof uses Cauchyâ€™s Integral theorem on the absolute value of the zeta function. Then, a check for convergence proves this result. Then the proof provides that if then has no zeros. In other words, is a zero-free region of the gamma function. This is proven by a contradiction, where it is assumed that a zero of the function exists on this region, but is contradicted by the isolated zeros property of holomorphic functions. We then prove that is a zero-free region for . This is done by writing as an exponential in terms of the Lambda function, if for some prime p and 0 otherwise, as

The proof uses properties of the zeta function and its zeros to simplify this expression, and by using a trigonometric identity it is proven that has no zeros for . The paper continues with another crucial result in the study of the zeta function, that is, the only zeros of the zeta function for are the negative even integers. This proof is carried out using the following functional Equation for the function.

The zeros of this function will occur at the poles of , which are at the negative even integers. Hence, it proves that the only zeros for are at the negative even integers, with imaginary part zero.

Section 5 comprises the contribution of this paper. This section presents a program the author wrote to compute the zeros of the zeta function. The program works by computing for . It then makes a list of all the outputs, and if the absolute value of the output is close to zero, then the value is counted as a zero. The program computes values of s for . There are a few uncertainties in the program, which are discussed briefly in the paper.

## Complex Analysis Background

This section discusses the necessary background information needed to understand and comprehend the results of this paper. It provides the knowledge required for holomorphic functions and Cauchyâ€™s Theorem, which pertain to contour integration. These are crucial to study the zeta function. These topics are explained in much more detail in^{2}

#### Holomorphic functions

To understand Cauchyâ€™s theorem, it is essential to understand what holomorphic functions are, since they are a pillar of Cauchyâ€™s Theorem. A function f is said to be holomorphic at a point in an open set in C, if the following limit converges

where with . A point to note is that because it may approach 0 from any direction, unlike on R where there are only two possible directions. The above equation is for a point on the function f, which means that if the limit exists then the function is said to be holomorphic at the point and not for all . A function f is said to be holomorphic if

exists on every point on the function f. The above limit is the derivative of a function which means that it can be expressed as

This implies that for an open set on and a complex-valued function f on , f is said to be a holomorphic function if the derivative exists on every point of the function f. This also means that the function is continuous and an intuitive way of defining a holomorphic function is that if a complex function f is smooth, then f is a holomorphic function.

**Few Examples**

\textbf{Example 1.} The function f(z) = is holomorphic for

Let F(z) =

Since the limit exists f is holomorphic.

Proof. Let f(z) =

#### Cauchyâ€™s Theorem and Its applications

This section will focus on Cauchyâ€™s Theorem and Cauchyâ€™s Integral formula, which are very important in the field of complex analysis and line integrals in the complex plane. Cauchyâ€™s theorem states that if f is a holomorphic function in an open set and is a closed curve (meaning that it has the same start and end point) in such that its interior is also contained in then

The above equation is very loosely stated because is a separate variable from the complex numbers. This problem may be proven straightforwardly by parametrized and f(z) so the integral is expressed in terms of a single variable: Define z : [a,b]

Now that the integral is in terms of one variable it becomes relatively easy to evaluate.

Even a basic proof of Cauchyâ€™s Theorem requires a fair bit of knowledge about closed contours which is why this paper will not look into the proof. Instead of proving the theorem, the paper will show that this equation does work by using it for special cases of f(z). The examples shown in â€˜Few Examplesâ€™ will be used as f, and for simplicity, the closed contour or will be the unit circle in both examples.

#### Few Examples

Example 3. The integral of f(z) = for over the unit circle is zero.

This problem can be solved by Cauchyâ€™s Theorem if it is proven that f is holomorphic. Since the unit circle is a closed contour. Since it is already proven that f is holomorphic it implies that the integral is equal to zero. This is why Cauchyâ€™s theorem is applicable in many cases, as it just requires knowing whether is closed and f is holomorphic. To check the validity of the method we evaluate the integral of f on , where f is and is the unit circle

Now,

This proves that the integral of f over is equal to zero and that Cauchyâ€™s Theorem is true for the special case

Example 4. Is the integral of f(z) = over the unit circle equal to zero?

Using Cauchyâ€™s theorem, we can say that the integral is not equal to zero because, as proven above, f is not holomorphic.

Now,

Example 5^{2}

(Hint: The Integral equals Use indented semicircle.)

Proof. We evaluate this Contour, over the function

We can further split into four separate integrals and evaluate them individually. Since f is holomorphic everywhere except at z = 0, and does not consist (0 + 0i), this is an integral over a closed contour over a holomorphic curve, which means that by Cauchyâ€™s Theorem

We now evaluate the first two integrals

Making the substitution and in the first integral

Since is a dummy variable, the integral above can be expressed as

If we take the limit of the integral as and we get the integral in the problem

Now we evaluate the integral over

Now

Finally, we evaluate the integral over

Now:

We note that

Since this integral is strictly positive and is greater than or equal to zero, by the squeeze theorem the value of the integral must be zero.

Now add the three the integrals

## The Reimann zeta function and the gamma function

The definition of the zeta function, for is

The Riemann zeta function can be continued analytically for . This function is the base for important observations such as the prime number theorem. This is a holomorphic function in the half plane and when continued it extends and produces several trivial zeros or roots, which are significant in many other fields â€“ not just mathematics. Finding the nontrivial zeros of the zeta function is an important problem in mathematics and it was proposed by Bernhard Reimann himself.

There are many proofs that the function can be analytically continued, or that the function can be extended graphically into the complex plane. See^{2} , page 182 for the proof. This section of the paper is about the representation of in terms of an integral, which can be used to compute values of (s) at different s.

Theorem 3.2. If , then the zeta function Ganbrexpossed as

where.

Proof: This proof modifies to produce and the equation isroarcanged to make the subject. In the definition of the gamma function,

making, the substitution , for , implies that ndu. Heace, we get:

Now taking the sum from zero to infinity on both sides while dividing both sides by :

Since the integral is strictly convergent and positive when applied on the given limits. Thus. the sum and integral can be swapped on the right-hand side. On the left-hand side is

independent of , which means that can be considered as a constant and factored out of the sum. Putting all the observations together yields:

Notice that the left-hand side of the equation is just , derived from the definition of . The infinite series on the right-hand side is a geometric progression, with Thus:

Therefore can be written as:

## Zeros of the zeta function

This section provides proofs regarding the zeros of the zeta function, where it proves that converges and has no zeros for has no zeros for and that only has zeros in . These proofs are important to study the zeta function and its zeros, and to study the critical strip and the critical line.

Proposition 4.1. If , then the series converges.

Proof:

By Cauchy’s Integral theorem, the above sum must converge or diverge if its corresponding integral converges or diverges.

For the above expression to converge , which implies that . Since , must converge when .

Theorem 4.2. then .

Proof. Assume as a contradiction that there exists such that and . Then for any :

Substituting , it follows that

Let such that ; such a exists by the integral definition of the function. Multiplying above equation by and integrating with respect to yields

Substituting :

Since is a holomorphic function for , must be an isolated zero

^{2}. On the other hand, the above argument shows that is also a zero for every , which implies that cannot be an isolated zero. Indeed, by choosing to be small enough, is a zero of which can be made arbitrarily close to . This contradicts the fact that is an isolated zero.

Proposition 4.4. The zeta function has no zeros for .

Proof. If , then

In the above equation, is the function defined by

This implies that the summand will be zero if is not a power of a prime, which yields,

The absolute value of \cite{mpmathâ€™s} is giver by

Using ,

Using the identity that , with implies

In particular, if .

Proposition 4.10. The only zeros of the zeta function with are the negative even integers .

Proof. The proof is based upon expressing as a functional equation, which is

Above Equation is derived in^{2}. For to be equal to zero, one of the three parts of the equations must equal zero. Note that cannot equal zero. Also, is not zero because, as proved above, has no zeros for . Further, is also never equal to zero. Hence, the zeros of with coincide with the poles of . has simple poles at the negative integers, including zero; see Section 6.1.2 of^{2}.

## Numerical Analysis of the zeros

This section of the paper presents a program the author wrote to compute the values of the zeta function on the critical line. The program provides a list of the non-trivial zeros of the zeta function by using the identities and theorems derived above.

#### Computing the zeros of

It is hypothesized that all the zeros of the zeta function lie on the Critical Line. In other words, the zeta function has zeros where the Real part of the input is .The program in Appendix A computes zeta on the critical strip. All it essentially does is that it computes for . Then the Real and Imaginary parts of the output are stored in different lists, namely, and . Then the program checks if the distance of the output from the origin is less than 0.01. If this condition is met then that specific output is added to another list, which is called zeros. The elements in zeros are the zeros of the zeta function. After which the program uses Manim^{3} to make a plot of the values obtained. A line graph is plotted, with the x values being the real part of the out and the y values being the imaginary part of the output. This Plot shows how the distribution of the zeros is along the critical line, and the zeta zeros are where the line intercepts the origin.

To calculate the values of the zeta function, the program uses a library called MpMath^{4}. The Library has inbuilt functions to calculate the zeta function and other functions related to it, such as the gamma function. The documentation provides detailed information about how it calculates these functions. The definitions used are in line with that shown in the paper, and thus this library was used. The program also uses NumPy^{5}, which is a library that provides various functions including the ones used in complex Analysis.

This is essentially the plot obtained after computing values for the zeta function on the critical line. The horizontal axis is the axis and the vertical axis is the axis. In the simulation above, a zeta-zero is essentially when the yellow line intercepts the origin. These values are noted down by the program and stored, the obtained values are listed in Appendix A. The images shown below are the same simulation, but the domain of the part is larger.

#### Uncertainties in the Program

The accuracy of the program is until 2 decimal places for the output of each input. Which is why some roots may be incorrect or repeated. A repeated root is two numbers that are in close vicinity to each other, which indicate the same root but due to the accuracy set in the program, the zero is listed as two different numbers. An example of this is 14.13 and 14.14, both these numbers are not roots of the zeta function, but are close approximations of the actual root, that lies somewhere between these two numbers.

The uncertainty of the program can be fixed by taking into consideration more decimal places when computing the values. The accuracy of the results will improve when the order of magnitude of the output becomes more accurate.

## Appendix A. Source code

zeros = [14.13, 14.14, 21.02, 21.03, 25.01, 30.42, 30.43,

32.93, 32.94, 37.59, 40.92, 43.33, 48.00, 48.01, 49.77,

49.78, 52.97, 56.45, 59.34, 59.35, 60.83, 65.11, 67.08,

69.55, 72.07, 75.70, 75.71, 77.14, 77.15, 79.34, 82.91,

84.74, 87.42, 87.43, 88.81, 92.49, 94.65, 95.87, 98.83,

101.32, 103.73, 105.45, 107.17, 111.03, 111.87, 111.88,

114.32, 118.79, 121.37, 122.95, 124.26, 129.58, 131.09,

133.50, 134.76, 139.74, 141.12, 143.11, 146.00, 147.42,

150.05, 150.92, 150.93, 156.11, 157.60, 158.85, 161.19,

163.03, 165.54, 169.09, 169.10, 169.91, 173.41, 174.75,

176.44, 178.38, 184.87, 184.88, 185.60, 187.23, 192.03,

193.08, 196.88, 202.49, 204.19, 211.69, 213.35, 214.55,

216.17, 219.07, 220.71, 220.72, 221.43, 224.01, 224.98,

227.42, 229.34, 231.25, 231.99, 237.77, 241.05, 242.82,

244.07, 247.14, 248.10, 249.57, 253.07, 256.38, 258.61,

259.87, 259.88, 265.56, 266.61, 266.62, 267.92, 269.97,

273.46, 275.59, 276.45, 278.25, 279.23, 283.21, 286.67,

287.91, 289.58, 293.56, 294.96, 294.97, 295.57, 297.98,

299.84, 301.65, 302.70, 304.86, 305.73, 307.22, 310.11,

312.43, 321.16, 322.14, 323.47, 329.03, 329.95, 333.65,

334.21, 338.34, 339.86, 341.04, 346.35, 347.27, 350.41,

351.88, 353.49, 356.02, 357.15, 357.95, 361.29, 363.33,

367.99, 368.97, 370.05, 373.06, 375.82, 375.83, 376.32,

376.33, 384.96, 385.86, 392.25, 393.43, 395.58, 396.38,

397.92, 401.84, 402.86, 404.24, 407.58, 411.97, 413.26,

415.02, 415.45, 415.46, 419.86, 420.64, 425.07, 427.21,

428.13, 430.33, 431.30, 432.14, 433.89, 436.16, 437.58,

438.62, 439.92, 444.32, 446.86, 447.44, 449.15, 450.13,

456.33, 459.51, 460.09, 465.67, 466.57, 467.44, 472.80,

475.60, 476.77, 478.94, 481.83, 483.85, 485.54, 486.53,

488.38, 489.66, 491.40, 493.31, 493.96, 495.36, 496.43,

498.58, 500.31, 501.60, 502.28, 504.50, 508.80, 511.56,

512.62, 513.67, 517.59, 522.46, 523.96, 527.90, 528.41,

529.81, 532.69, 533.78, 537.07, 538.43, 540.21, 540.22,

540.63, 540.64, 547.01, 547.93, 550.97, 552.05, 555.79,

556.90, 557.56, 557.57, 560.24, 562.56, 564.16, 564.50,

564.51, 566.70, 567.73, 570.05, 572.42, 575.09, 575.81,

577.04, 579.10, 584.56, 586.74, 588.14, 590.66, 591.73,

592.57, 595.73, 596.36, 601.60, 602.58, 603.63, 609.39,

610.84, 613.60, 615.54, 620.27, 621.71, 624.27, 626.02,

627.27, 630.47, 630.48, 630.80, 630.81, 637.40, 637.92,

637.93, 638.93, 643.28, 644.99, 646.35, 647.76, 648.79,

650.20, 650.67, 653.65, 654.30, 655.71, 665.34, 666.51,

666.52, 667.15, 672.46, 673.04, 676.14, 677.23, 677.80,

682.60, 684.01, 684.97, 689.37, 692.45, 693.18, 700.30,

701.30, 702.23, 708.27, 709.23, 711.13, 711.90, 712.75,

718.74, 721.35, 722.28, 728.40, 728.41, 728.76, 734.79,

738.58, 739.91, 740.57, 746.50, 747.67, 747.68, 748.24,

750.66, 750.97, 755.84, 756.77, 758.10, 758.90, 762.70,

763.59, 764.31, 767.22, 768.28, 771.07, 772.96, 774.12,

775.05, 776.00, 777.30, 780.35, 782.14, 782.60, 784.29,

785.74, 786.46, 787.47, 790.06, 790.83, 792.43, 792.89,

802.54, 803.24, 809.20, 810.08, 812.77, 814.87, 818.38,

820.72, 821.71, 821.72, 822.20, 826.04, 828.34, 831.80,

836.69, 837.35, 838.25, 842.04, 847.97, 848.49, 849.86,

857.31, 860.41, 861.17, 863.19, 864.34, 866.42, 868.67,

872.19, 873.10, 873.91, 876.60, 879.38, 883.43, 884.20,

885.27, 893.12, 894.89, 895.40, 899.22, 899.86, 900.85,

902.24, 903.10, 905.83, 912.33, 912.82, 914.73, 918.84,

919.45, 922.50, 927.85, 928.66, 931.01, 936.23, 939.66,

946.76, 946.77, 947.07, 947.08, 950.15, 954.13, 954.83,

957.51, 959.46, 961.67, 963.18, 963.57, 966.11, 967.37,

971.07, 973.87, 976.18, 980.58, 981.29, 986.13, 992.73,

993.21, 993.22, 999.79]

## References

- G. H. Hardy and J. E. Littlewood, Mathematische Zeitschrift, 1921, 10, 283â€“317. [↩]
- E. M.Stein, Complex Analysis, 2003. [↩] [↩] [↩] [↩] [↩] [↩] [↩] [↩]
- Manim Community Edition, https://docs.manim.community/index.html. [↩]
- Welcome to mpmathâ€™s documentation! â€” mpmath 1.3.0 documentation, https://mpmath.org/doc/current/. [↩]
- NumPy Documentation, https://numpy.org/doc/. [↩]