What is zeta in math

The Tale of Riemann

Many acclaimed mathematicians and scientists, Georg Friedrich Bernhard Riemann

and Albert Einstein among many others, have come from humble and trying

backgrounds. I, while certainly not comparing myself to those icons, also come from a

humble background. Math and science are of great interest to me, especially math.

Riemann and I are both shy and delve into math as a haven. Riemann provided society

with the Riemann Hypothesis, a problem that continues to vex mathematicians over a

century and a half later. It is one of Clay Mathematics Institute’s Millennium Problems,

and while progress has been made, no one has been able to prove the theorem yet. This

paper will explore how Riemann overcame his hard beginnings to become one of the

greatest mathematicians of all time.

Riemann was born on September 17, 1826 in Breselenz, a village near Dannenberg in

the Kingdom of Hanover. His father, Friedrich Bernhard Riemann, was a Lutheran

minister and a veteran of the Napoleonic Wars. “The flat, damp countryside; the

draughty house lit only by oil lamps and candles, ill-heated in winter and ill-ventilated in

summer; long spells of sickness among siblings who themselves were never quite well

(they seem all to have suffered from tuberculosis); the tiny and monotonous social round

of a parson’s family in a remote village; the inadequate and unbalanced diet on the stodgy

side of a stodgy national cuisine” (Derbyshire).

As an impoverished family with a menial income, the children often suffered from

malnutrition. On top of their economic situation, Riemann also suffered from anxiety

and depression and he only felt at ease when he was at home. These modest

accommodations would serve as the center of Riemann’s emotional world for most of his

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life. Despite all these hardships, the Riemann family was a close-knit and loving pious

family.

Riemann was homeschooled by his father until he was 14 years old. When his father

had saved enough money, Riemann moved to Hanover to live with his grandmother and

finally attend formal school. He was very shy and extremely homesick while he was

there. When she died two years later, he attended school closer to home. He was not a

good scholar, only doing well in what he found interesting, mathematics mostly. He

thought deeply about philosophy and saw all his mathematical work in a larger

philosophical context. He was often late turning in assignments because he was a

perfectionist and desired turning in only flawless compositions. The school director

arranged for Riemann to board with a Hebrew teacher named Seffer. (Upon Riemann’s

death, Seffer said “I learnt more from him, than he from me” (Haftendorn).) Under

Seffer’s direction, Riemann improved drastically academically. So much so that in 1846

he was admitted into the University of Gottingen as a student of theology. Riemann had

always hoped to follow in his father’s footsteps.

It was at Gottingen that Riemann’s life would take a sharp turn. He was close to home

and his family, but more importantly, he attended lectures on linear algebra given by Carl

Friedrich Gauss, the greatest mathematician of his time and theory of equations lectures

by Moritz Stern. Riemann became so infatuated with mathematics that he and his father

decided it was better for him to have a career as a mathematician instead of pursuing the

ministry. Riemann received his doctorate in 1851, at age 25, having submitted a

dissertation on complex function theory. He stayed at Gottingen first as a lecturer, then

he became an associate professor in 1857.

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Riemann had left Gottingen to attend The Berlin Academy for a year in 1848. The

Berlin Academy was a pure research institute modeled after England’s Royal Society. In

1859, Bernhard Riemann was appointed a corresponding member of the Berlin Academy.

The Berlin Academy did so on the merits of Riemann’s 1851 dissertation and his 1857

work on Abelian functions. It was a tremendous honor for Riemann to be elected a

member of the Berlin Academy at the age of 32. Upon his appointment, as was the

custom for all appointments, Riemann had to submit an original paper describing his

research to the Academy. Riemann submitted “Uber die Anzahl der Primzahlen unter

einer gegebenen Grosse” or “On the Number of Prime Numbers Less Than A Given

Quantity”. Mathematics has not been quite the same since (Derbyshire).

The Reimann Hypothesis is a fundamental mathematical conjecture that has huge

implications for the rest of math and number theory. The Riemann Hypothesis states that

when the Riemann Zeta Function crosses zero (except for the zeros between -10 and 0),

the real part of the complex number is one-half. That's a rather abstract mathematical

statement having to do with what numbers you can put into a particular mathematical

function that will make it equal zero.

Riemann observed that the frequency of prime numbers is very closely related to the

behavior of an elaborate function. Riemann showed that the distribution of prime

numbers seems to be profoundly linked to the non-trivial zeros of the zeta function,

where the complex number’s real part equals one-half. The function below:

ζ(s) = 1 + 1

2𝑠 +

1

3𝑠 +

1

4𝑠 + ⋯

is called the Riemann Zeta Function.

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The Riemann Hypothesis asserts that all interesting solutions of the equation:

ζ(s) = 0 lie on a certain vertical straight line.

The Riemann Zeta Function inputs and outputs complex number values. The inputs

that give the output zero are called zeros of the zeta function. Many zeros have been

found. The "obvious" ones to find are the negative even integers. This follows from

Riemann's functional equation. The hypothesis states the real part of any non-trivial zero

of the Riemann Zeta Function is one-half.

Riemann showed that the distribution of prime numbers seems to be profoundly linked

to the non-trivial zeros of the zeta function, where the complex number’s real part equals

one-half. Riemann described the formula for the prime counting function as:

𝜋(x) = Approximation term + Error term

The approximation term is the same presently as the one present in the existing literature

of the time. Riemann provided an expression for the error term. The error term involves

all the points at which the Riemann Zeta Function assumes the value 0. Hence, the

location of the zeros of the Riemann Zeta Function are crucial. It has been proved that

there are infinitely many zeros of the Riemann Zeta Function in the critical strip (real

part between 0 and 1) indicated in Figure 1.

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Figure 1: The critical strip is the real part between 0 and 1 where infinitely many zeros of the Riemann Zeta

Function are located.

Finally, the Riemann Hypothesis states that all the zeros lie on the critical line (shown

in black) on the real part of one-half. Proving this hypothesis has eluded mathematicians

for a century now (Bhattcharya).

Numerous attempts have been made to prove the Riemann Hypothesis. It was listed

on both German mathematician David Hilbert’s 1900 list of 23 mathematical problems

and the Clay Mathematics Institute’s Millennium Problems List. The ramifications of

proving Riemann lie in many areas of mathematics, number theory and string theory that

rely on this being correct. Many other theories and hypotheses would fail if they were

reliant on the Riemann Hypothesis being true.

http://www.freelunch.co.in/wp-content/uploads/2017/06/critical-strip.png
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One such approach came in 1931 when Kurt Godel presented the Godel

Incompleteness Theorem. It states that not every theorem arising from a certain set of

axioms can be proved even if the said theorem is true. Axioms are the foundation of

mathematics, and in our case, the foundations of Real Analysis and Set Theory serve as

our axioms. In this light, it is possible that the Riemann Hypothesis cannot be proved

using our present foundation of mathematics, even if it is true.

In 1927, Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity

of Jensen Polynomials for the Riemann Zeta Function 𝜁(𝑠) at its point of symmetry. This

is only true if all the Jensen Polynomials associated with the Riemann Zeta Function have

only zeros that are real. This means the values for which the polynomial equals zero are

not imaginary numbers. However, there are also infinitely many of these Jensen

Polynomials and requires a similar breadth (Conover). The most recent progress in

proving the Riemann Hypothesis occurred in May of 2019 by Ken Ono, a number

theorist at Emory University, who was working on the “hyperbolicity of Jensen

polynomials” (Letzter).

In a paper published in the journal Proceedings of the Natural Academy of Sciences

(PNAS), Ono and his colleagues proved that in many, many cases the zeros of the Jensen

Polynomials are only real. Unfortunately, there remain some cases where they don’t

know definitively if the zeros are only real. Ono goes on to say that it is unclear how far

away a more advanced proof is that would definitively show the criterion is true in all

cases, and thus finally proving the Riemann Hypothesis (Letzter).

Ono may have developed an important underlying framework for solving the Riemann

Hypothesis. “Although this remains far away from proving the Riemann hypothesis, it is

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a big step forward” Encrico Bombieri, a Princeton number theorist who won a Field’s

Medal in 1974 for his work related to the Riemann Hypothesis, wrote in an

accompanying May 23 PNAS article. "There is no doubt that this paper will inspire

further fundamental work in other areas of number theory as well as in mathematical

physics." (Letzter) I am inspired by Ono’s and others work on proving the Riemann

Hypothesis and would be interested in further examining the criterion.

Riemann is a fascinating figure in mathematics and number theory. If one only

considered his outside appearance and demeanor, he might be considered pathetic, sad,

shy and depressed, as Riemann was largely withdrawn in his everyday life. Riemann is

like the character Charles Strickland in Somerset Maugham’s novel, The Moon and

Sixpence. Strickland is an artist with a dull exterior that conceals the soul of a genius.

He is poor and sick and lives for art, as Riemann did for mathematics, no matter what the

price was to himself. Strickland moves to Tahiti to pursue his vision of art, even though

he is going blind and has leprosy. When a local doctor checks on him, he is in a shabby

dilapidated hut. However, when the doctor goes inside, he finds the walls covered from

floor to ceiling with brilliant paintings. As with that hut, so it was with Riemann.

Outwardly, he was pitiable; inwardly though, he burned brighter than the sun

(Derbyshire).

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Bibliography

Bhattacharya, Anwesh. “The Riemann Zeta Hypothesis.” The Riemann Zeta Hypothesis,

25 June 2017, www.freelunch.co.in/the-riemann-zeta-hypothesis/.

Conover, Emily. “Mathematicians Report Possible Progress on the Riemann Hypothesis.”

Science News, 8 Aug. 2019, www.sciencenews.org/article/mathematicians-

progress-riemann-hypothesis-proof?signup=success.

Derbyshire, John. Prime Obsession. Joseph Henry Press, 2003.

Haftendorn, Dörte. “Bernhard Riemann.” Johanneum Lüneburg Bernhard Riemann, Dec.

1996, web.archive.org/web/20050903035028/ www.fh-

lueneburg.de/u1/gym03/englpage/chronik/riemann/riemann.htm.

Letzter, Rafi. “Mathematicians Edge Closer to Solving a 'Million Dollar' Math Problem.”

LiveScience, Purch, 28 May 2019, www.livescience.com/65577-riemann-

hypothesis-big-step-math.html.

http://www.freelunch.co.in/the-riemann-zeta-hypothesis/
http://www.sciencenews.org/article/mathematicians-progress-riemann-hypothesis-proof?signup=success
http://www.sciencenews.org/article/mathematicians-progress-riemann-hypothesis-proof?signup=success
http://www.fh-lueneburg.de/u1/gym03/englpage/chronik/riemann/riemann.htm
http://www.fh-lueneburg.de/u1/gym03/englpage/chronik/riemann/riemann.htm
http://www.livescience.com/65577-riemann-hypothesis-big-step-math.html
http://www.livescience.com/65577-riemann-hypothesis-big-step-math.html