Western Kentucky University TopSCHOLAR® Masters Theses & Specialist Projects Graduate School 8-2011 Higher Derivatives of the Hurwitz Zeta Function Jason Musser Western Kentucky University, [email protected] Follow this and additional works at: http://digitalcommons.wku.edu/theses Part of the Number Theory Commons Recommended Citation Musser, Jason, "Higher Derivatives of the Hurwitz Zeta Function" (2011). Masters Theses & Specialist Projects. Paper 1093. http://digitalcommons.wku.edu/theses/1093 This Thesis is brought to you for free and open access by TopSCHOLAR®. It has been accepted for inclusion in Masters Theses & Specialist Projects by an authorized administrator of TopSCHOLAR®. For more information, please contact [email protected]. HIGHER DERIVATIVES OF THE HURWITZ ZETA FUNCTION A Thesis Presented to The Faculty of the Department of Mathematics Western Kentucky University Bowling Green, Kentucky In Partial Fulfillment Of the Requirements for the Degree Master of Science By Jason Musser August 2011 ACKNOWLEDGMENTS I would like to thank Dr. Dominic Lanphier for his insightful guidance in this endeavor. iii TABLE OF CONTENTS Abstract v Chapter 1 The Riemann Zeta and Hurwitz Zeta Functions 1.1 Introduction 1 1.2 The Gamma Function 2 1.3 The Riemann Zeta Function 3 1.4 The Hurwitz Zeta Function 5 Chapter 2 The Stieltjes Constants and Euler's Constant 2.1 The Stieltjes Constants 8 2.2 Euler's Constant 9 Chapter 3 Higher Derivatives of the Riemann Zeta Function 3.1 Series Expansion of the Functional Equation 11 Chapter 4 Higher Derivatives of the Hurwitz Zeta Function 4.1 Series Expansion of the Functional Equation 17 4.2 Higher Derivatives of the Hurwitz Zeta Function 21 4.3 Main Results 26 4.4 Applications to the Stieltjes Constants 28 4.5 Examples 29 iv Chapter 5 Numerical Results 5.1 Numerical Results 32 Appendix 39 References 47 v HIGHER DERIVATIVES OF THE HURWITZ ZETA FUNCTION Jason Musser August 2011 46 Pages Directed by: Dominic Lanphier, Tilak Bhattacharya, Claus Ernst Department of Mathematics Western Kentucky University The Riemann zeta function ζ(s) is one of the most fundamental functions in number theory. Euler demonstrated that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have been studied by several mathematicians. Apostol in particular gave a computable formula for the values of the derivatives of ζ(s) at s = 0. The Hurwitz zeta function ζ(s; q) is a generalization of ζ(s). We modify Apostol's methods to find values of the derivatives of ζ(s; q) with respect to s at s = 0. As a consequence, we obtain relations among certain important constants, the generalized Stieltjes constants. We also give numerical estimates of several values of the derivatives of ζ(s; q). vi Chapter 1 The Riemann Zeta and Hurwitz Zeta Functions In this chapter an introduction, definitions of the gamma function, the Riemann zeta function, and the Hurwitz zeta function are given. Various properties of each function including integral representations that give their meromorphic continuation to C which can be found in [1] and [6] are also shown. This is done in preparation for ultimately discussing the derivatives of the Hurwitz zeta function. Many of these properties can be found in any introductory analytic number theory text. 1.1 Introduction In 1859 Bernhard Riemann discussed the significance of the Riemann zeta function ζ(s) in his paper Uber¨ die Anzahl der Primzahlen unter einer gegebenen Gr¨osse (On the Number of Primes Less Than a Given Magnitude). He outlined a method whereby one can study prime numbers from the analytic properties of ζ(s). These ideas culminated in a proof of the celebrated prime number theorem in 1900 by Hadamard and de la Vall´eePoussin, which gives an asymptotic formula for the number of primes less than a given value. In Riemann's original paper he stated a conjecture that is now famously known as the Riemann hypothesis. The critical strip of the Riemann zeta function is the set s 2 C so that 0 < Re(s) < 1. From properties of ζ(s) we know that ζ(s) 6= 0 for Re(s) > 1. The analytic continuation of ζ(s) allows us to define ζ(s) for any s 2 C, s 6= 1. It can be shown that ζ(s) 6= 0 for Re(s) ≥ 1 and this result implies the Prime 1 Number Theorem. It is known that there are values σ0 = 2 + it so that ζ(σ0) = 0. The Riemann hypothesis asserts that all nontrivial zeros of ζ(s) lie on the vertical 1 line at Re(s) = 2. Therefore, the Riemann hypothesis asserts that ζ(s) 6= 0 for 1 Re(s) > 2. Although it has not been shown to be true or false, many zeros lie on this line. One of the reasons the Riemann hypothesis is important is because given 1 that the Riemann hypothesis is true, further very powerful deductions can be made about the distribution of prime numbers. To further study the Riemann zeta function, Adolf Hurwitz defined a more generalized form of the Riemann zeta function refered to as the Hurwitz zeta function ζ(s; q). Although the relationship to the prime numbers is lost in the generalization, insight can be gained on the Riemann zeta function by studying the Hurwitz zeta function due to relationships between the two. One of the properties that has been studied is the derivatives of the Riemann zeta function at s = 0 [2]. As with ζ(s), the Hurwitz zeta function has an analytic continuation to all s 2 C, s 6= 1. Thus ζ(s; q) is analytic at s = 0. The Taylor series at s = 0 has a radius of convergence 1. In this thesis we find the derivatives at s = 0 of the Hurwitz zeta function by modifying methods used to study ζ(s). Some numerical results are also found. For all instances that refer to Mathematica the ` software Mathematica 7:0 was used. Note that in all chapters f(`)(s) = d f is ds` defined for any meromorphic f. 1.2 The Gamma Function The gamma function Γ(n) is defined as follows from Chapter one of [6] : For n 2 Z, n > 0, Γ(n) = (n − 1)!; n = 1; 2;::: and 0! = 1: The gamma function has an integral representation from chapter one of [6] given by Z 1 Γ(s) = ts−1e−tdt 0 that converges absolutely for Re(s) > 0 and allows a definition of Γ(s) for all s 2 C. The integral converges for all complex s with s 6= 0; −1; −2; ··· . The gamma function has simple poles at these points. 2 Some selected properties of the gamma function follow. Three formulas that are functional equations of the gamma function are given. From integration by parts the recursion formula can be obtained ((1.1.6) in [6]) Γ(s + 1) = sΓ(s) and Γ(n) = (n − 1)!; n = 1; 2;::: Euler's reflection formula ((1.2.1) in [6]) states π Γ(1 − n)Γ(n) = : sin(πn) The duplication formula ((1.5.1) in [6]) states 1 1 Γ(z)Γ z + = 21−2z Γ Γ(2z) 2 2 A well known value of the gamma function ((2.14) in [3]) is 1 p (1) Γ = π: 2 1.3 The Riemann Zeta Function The Riemann zeta function ζ(s) as found in Chapter twelve of [1] is defined by the series 1 X 1 ζ(s) = ns n=1 for Re(s) > 1. It has an analytic continuation to all s = σ + it except for a pole at s = 1 with residue 1 as found in the introduction of [10], and it can be obtained from the integral representation Γ(1 − s) Z (−z)s−1 ζ(s) = − z dz: 2πi C e − 1 3 The contour C starts at infinity on the positive real axis, encircles the origin once, excluding the points ±2πi; ±4πi; : : : and returns to where it begins. Γ(s) is the gamma function. Also, if Re(s) > 1 then we can express the product of the gamma function and Riemann zeta function as found in the introduction of [10] by Z 1 xs−1 Γ(s)ζ(s) = x dx 0 e − 1 In the introduction of [10], Euler found that for Re(s) > 1, 1 X 1 Y 1 = : ns 1 − p−s n=1 8 primes p This shows an interesting relationship between the Riemann zeta function and the prime numbers. Like the gamma function, the Riemann zeta function obeys certain formulas. For example, the Riemann zeta function as found in Chapter twelve of [1] has a functional equation πs (2) ζ(s) = 2(2π)s−1sin Γ(1 − s)ζ(1 − s); 8s 2 : 2 C Many values of the Riemann zeta function are particularly interesting. Values of the Riemann zeta function have a connection with the Bernoulli numbers. The Bernoulli numbers Bm are the coefficients of the following series expansion ((1.2.10)in [6]) 1 m t X Bmt (3) = et − 1 m! m=0 1 1 1 1 for t 2 R and t 6= 0. For example B0 = 1, B1 = 2, B2 = 6, B4 = −30, B6 = 42, 4 1 5 B8 = −30, B10 = 66. If k is an even and positive integer we have ((1.2.11)in [6]) k B (2π)k ζ(k) = (−1) 2 +1 k (2k)! As examples, this equation gives 1 X 1 π2 ζ(2) = = n2 6 n=1 and 1 X 1 π4 ζ(4) = = : n4 90 n=1 These results were first discovered by Euler.