Meromorphic Dirichlet Series by Corey Everlove A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2018 Doctoral Committee: Professor Jeffrey C. Lagarias, Chair Professor Deborah E. Goldberg Associate Professor Sarah C. Koch Professor Hugh L. Montgomery Professor Karen E. Smith Corey Everlove [email protected] ORCID id: 0000-0001-6301-3085 Acknowledgements There are so many people who I’d like to thank—those who made my time in graduate school here at Michigan such a great experience and those who made this work possible. Thanks to Paul, Angela, Fernando, Gavin, Steven, and everyone else who made teaching at Michigan such a pleasant and rewarding experience. Thanks to Deborah, Shannon, Tony, and everyone at M-Sci for six great summers of teaching. Thanks to the many friends I’ve made here at Michigan, far too numerous to list. Thanks to my parents for always encouraging me to pursue my varied academic interests over the years. Thanks to Salman for your emotional support when work was overwhelming. And thanks to my cats, Link and Luna, for keeping me company while I worked. But most of all, I would like to thank my advisor, Jeff Lagarias. Thank you for suggesting good problems, for many interesting mathematical discussions, and for all your advice on research, writing, and staying motivated. The work contained in this thesis was partially supported by NSF grants DMS-1401224 and DMS-1701576. ii Contents Acknowledgements ii List of Figures v Abstract vi Chapter 1: Introduction 1 1.1 Dirichlet series and meromorphic continuation . 1 1.2 Outline of the thesis and summary of results . 5 Chapter 2: Rings of Dirichlet series 14 2.1 Formal Dirichlet series . 14 2.2 Convergent Dirichlet series . 19 2.3 Meromorphic Dirichlet series . 24 2.4 Finite-order meromorphic Dirichlet series . 26 Chapter 3: Dirichlet series and additive convolutions 31 3.1 Introduction . 31 3.2 A relation between power series and Dirichlet series . 34 3.3 Dirichlet series associated to additive convolutions . 36 3.4 Dirichlet series and the calculus of finite differences . 40 3.5 Examples . 45 Chapter 4: Dirichlet series associated to digital sums 54 4.1 Introduction . 54 4.2 Sum-of-digits Dirichlet series . 59 4.3 Meromorphic continuation of Fb¹sº and Gb¹sº . 62 4.4 Meromorphic continuation of Dirichlet series for non-integer bases . 65 iii Chapter 5: Complex Ramanujan sums and interpolation of the sum-of-divisors function 72 5.1 Introduction . 72 5.2 Classical Ramanujan sums . 77 5.3 Complex Ramanujan sums . 78 5.4 The Dirichlet series generating function of the complex Ramanujan sums . 79 5.5 Interpolation of Fourier coefficients of Eisenstein series . 86 Appendix 90 Bibliography 93 iv List of Figures 1.1 A rectangular contour for using Perron’s formula. 3 4.1 A plot of Sβ¹10º for 1 ≤ β ≤ 15, using terms with jkj ≤ 1000 in the Fourier series for hβ¹xº. ....................................... 66 4.2 A plot of hβ¹2º for 1 ≤ β ≤ 8, using terms with jkj ≤ 1000 in the Fourier series for hβ¹xº. ......................................... 67 4.3 A plot of hβ¹log 2/log βº for 1 ≤ β ≤ 8, using terms with jkj ≤ 1000 in the Fourier series for hβ¹xº..................................... 67 v Abstract This thesis studies several problems concerning the meromorphic continuation of Dirichlet series to the complex plane. Í −s We show that if a Dirichlet series f ¹sº = ann has a meromorphic continuation to Í n the complex plane and the power series generating function bnz of a sequence bn has a meromorphic continuation to z = 1, then the Dirichlet series 1 Õ Õ −s aibj n n=1 i+j=n has a meromorphic continuation to the complex plane. Using specific choices of the sequence bn, we show that the Dirichlet series whose coefficients are a shift, forward or backward difference, or partial sum of the coefficients of f ¹sº has a meromorphic continuation to the complex plane. We study several examples of such Dirichlet series involving important arithmetic functions, including the Dirichlet series whose coefficients are the Chebyshev function, the Mertens function, the partial sums of the divisor function, or the partial sums of a Dirichlet character. We apply these results to study the Dirichlet series whose coefficients are the sum of the base-b digits of the integers. We also study the Dirichlet series whose coefficients are the cumulative sum of the base-b digits of the integers less than n. We show that these Dirichlet series have a meromorphic continuation to the complex plane, and we give the locations of the poles and the residue at each pole. We also consider an interpolation of the sum-of-digits and cumulative sum-of-digits functions from integer bases b ≥ 2 to a real parameter β > 1, and show that Dirichlet series attached to these interpolated sum-of-digits functions meromorphically continue one unit left of their halfplanes of convergence. Finally, we consider a one-parameter family of Dirichlet series related to Ramanujan sums. The classical Ramanujan sum cn¹mº is a function of two integer variables; we replace the integer parameter m with a complex number and consider the Dirichlet series attached to such complex Ramanujan sums. We show that this Dirichlet series continues to a meromorphic function of two complex variables and locate its singularities. vi Chapter 1 Introduction 1.1 Dirichlet series and meromorphic continuation This thesis studies some topics related to the meromorphic continuation of Dirichlet series. A Dirichlet series is a series of the form 1 Õ an f ¹sº = (1.1) ns n=1 with coefficients an 2 C, considered as a function of a complex variable s. As we review in Chapter 2, the region of convergence of a Dirichlet series is a halfplane: each Dirichlet series has an abscissa of convergence σc (possibly 1 or −∞) such that the series converges to a holomorphic function if Re¹sº > σc and diverges if Re¹sº < σc. The holomorphic function defined by a Dirichlet series in its halfplane of convergence may or may not analytically continue to a meromorphic function on a larger region of the complex plane. 1.1.1 The Riemann zeta function The prototypical example of a Dirichlet series is the Riemann zeta function defined for Re¹sº > 1 by the series 1 Õ 1 ζ¹sº = : (1.2) ns n=1 In 1737, Euler studied ζ¹sº as a function of a real variable. By writing the zeta function as a product over the prime numbers Ö 1 ζ¹sº = (1.3) − p−s p 1 and using the fact that ζ¹sº ! 1 as s ! 1+, Euler deduced that the sum Í 1/p of the reciprocals of the primes diverges. Here we see the beginning of a major theme of analytic number theory: 1 information about the analytic behavior of a Dirichlet series (the growth of ζ¹sº at s = 1) is used to deduce arithmetic information (a statement about the distribution of prime numbers). In 1859, Riemann studied the function ζ¹sº as a function of a complex variable (see the appendix to [18] for an English translation of Riemann’s memoir). Riemann proved that ζ¹sº, originally defined for Re¹sº > 1, continues to a meromorphic function on C with only singularity a simple pole at s = 1 with residue 1. Riemann also proved the functional equation s 1 − s π−s/2Γ ζ¹sº = π−(1−s)/2Γ ζ¹1 − sº (1.4) 2 2 relating the values of ζ¹sº for Re¹sº > 1/2 and Re¹sº < 1/2. From the functional equation and the nonvanishing of ζ¹sº for Re¹sº > 1, Riemann deduced that aside from the “trivial zeros” at s = −2k for integers k ≥ 1, the zeros of ζ¹sº must lie in the strip 0 ≤ Re¹sº ≤ 1. Riemann famously conjectured, based on some computational evidence, that all of the zeros in this strip must have real part 1/2. Riemann made some further conjectures about ζ¹sº and its relation to the distribution of the prime numbers; before stating a form of his conjectured “explicit formula”, we return to more general Dirichlet series. 1.1.2 Perron’s formula The following formula provides one of the important links between the analytic behavior of a Dirichlet series and information about its coefficients. Proposition 1.1.1 (Perron’s formula, see [52, sec. 9.42]). If 1 Õ an f ¹sº = (1.5) ns n=1 is a Dirichlet series convergent in the halfplane Re¹sº > σc, then for c > max¹σc; 0º, Õ0 1 ¹ c+iT xs an = lim f ¹sº ds; (1.6) T!1 πi s n≤x 2 c−iT where if x is an integer, the term ax in the sum on the left is counted with weight 1/2. More detailed versions of this relation bound the error in truncating the integral at height T instead of taking the limit or modify the factor xs/s in the integral to introduce weights in the sum of the coefficents an. If the Dirichlet series f ¹sº has a meromorphic continuation to the complex plane, then one might hope to use information about the singularities of f ¹sº to evaluate or estimate the 2 Im iT R Re L c −iT Figure 1.1: A rectangular contour for using Perron’s formula. integral on the right of (1.6) using the residue theorem. For example, one could integrate over a rectangular contour R as in Figure 1.1 by the residue theorem, then estimate the contributions of the integral over the top, left, and bottom sides.