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Riemann Zeta Function

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Contents:

  1. What is the Riemann Zeta Function?
  2. Hurwitz Zeta Function

What is the Riemann Zeta Function?

The Riemann Zeta Function is a generalized harmonic series, given by the formula:

Riemann Zeta Function

Riemann’s Zeta Function. Image: Wikimedia Commons.


Where n and s are two real numbers.

  • When n = 1, the zeta function is the same as the harmonic series. Like the harmonic series, it also diverges (fails to tend towards a certain number).
  • For all other values of n, the series conditionally converges (tends toward a number).

The function is useful in several areas, including properties of prime numbers and definite integrals.

Zeros of the Riemann Zeta Function

Zeros of a function are any input (i.e. any “x”) that results in the function equaling zero. For a basic function like y = 2(x), this is fairly easy to do, but it gets a little more complicated with the Riemann Zeta Function, mostly because it involves complex numbers. The even integers -2, -4, -6… are sometimes called the “trivial zeros” of the Riemann zeta function. However, other zeroes do exist and they are complex, “non-trivial” zeros. If you’re interested, Andrew Odlyzko has an extensive list, including the first 100 zeros (accurate to 1,000 decimal places). Here’s the first zero:

14.134725141734693790457251983562470270784257115699243175685567460149963429809256764949010393171561012779202971548797436766142691469882254582505363239447137780413381237205970549621955865860200555566725836010773700205410982661507542780517442591306254481978651072304938725629738321577420395215725674809332140034990468034346267314420920377385487141378317356396995365428113079680531491688529067820822980492643386667346233200787587617920056048680543568014444246510655975686659032286865105448594443206240727270320942745222130487487209241238514183514605427901524478338354254533440044879368067616973008190007313938549837362150130451672696838920039176285123212854220523969133425832275335164060169763527563758969537674920336127209259991730427075683087951184453489180086300826483125169112710682910523759617977431815170713545316775495153828937849036474709727019948485532209253574357909226125247736595518016975233461213977316005354125926747455725877801472609830808978600712532087509395997966660675378381214891908864977277554420656532052405

History

Leonhard Euler was the first to study the function, but Bernhard Riemann was the first to study it extensively. Riemann was preoccupied with complex analysis and introduced the Zeta Function in during his study of primes; perhaps his most significant contribution is his consideration of the zeta-function as a type of analytic function (Patterson, 1995).

The Riemann Hypothesis

Although many parts of the function have been investigated and proved, the Riemann hypothesis (simply stated as “The real part of every non-trivial zero of the Riemann zeta function is ½”) remains famously unproven.

Hurwitz Zeta Function

The “classical” Hurwitz zeta function (also called the generalized zeta function) is usually denoted by ζ (s, α) and is a generalization of the Riemann zeta function. The function, named after 19th century mathematician Adolf Hurwitz, can be defined for complex-numbered inputs s and a (with Re(a) > 0 and Re(s) > 1) as the series (Xu, 2018):
huriz zeta definition

The series is absolutely convergent for given values of a and a; It can also be extended to a meromorphic function for all s ≠ 1. When continued to the entire s-plane (the complex plane on which Laplace transforms are graphed), there is a simple pole at s = 1.


Historical Note

The function can really only be understood by reading it in its historical context (including Riemann’s developments). According to legend, Hurwitz spent so much time pondering the Riemann hypothesis that his copy of Riemann’s collected works (obtained from his library after his death) automatically fell open to the paged where Riemann’s hypothesis was stated. For a great introduction, I recommend Harold Edward’s book Riemann’s Zeta Function; It’s an older text but an easy read compared to some of the more esoteric texts on the zeta function.

Different Forms of the Hurwitz Zeta Function

This complex and interesting function has a plethora of different forms; It can be written as a sum of Dirichlet L-functions, as a series in terms of the gamma function, or in terms of the polygamma function (for positive integer inputs). When ζ(s, 1), the function becomes the Riemann zeta function.

Various extended forms of the function exist; A fairly simple version is the n-ple (“multiple”) Hurwitz zeta function, defined as (Srivastava & Choi, 2012):
hurwitz zeta function

Applications of the Hurwitz Zeta Function

The Hurwitz Zeta Function function has a wide variety of applications in mathematics and quantum mechanics (Xu, 2018), including:


  • Finding numerical prime density estimates (e.g. in the prime number theorem),
  • Calculating the Casimir effect in quantum mechanics,
  • Finding the pair production rate of Dirac particles.

References

Elizalde, E. et al. (1994). Zeta Regularization Techniques with Applications. World Scientific.
Patterson, S. (1995). An Introduction to the Theory of the Riemann Zeta-Function. Cambridge University Press.Srivastava, H. & Choi, J. (2012). Zeta and Q-Zeta Functions and Associated Series and Integrals. Elsevier Science.
Xu, A. (2018). Approximating the Hurwitz Zeta Function. Retrieved July 20, 2020 from: https://amzn.to/32ByCAn

CITE THIS AS:
Stephanie Glen. "Riemann Zeta Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/riemann-zeta-function/
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