254A, Notes 1: Elementary multiplicative number theory

Originally posted on What's new:

In analytic number theory, an arithmetic function is simply a function $latex {f: {bf N} rightarrow {bf C}}&fg=000000$ from the natural numbers $latex {{bf N} = {1,2,3,dots}}&fg=000000$ to the real or complex numbers. (One occasionally also considers arithmetic functions taking values in more general rings than $latex {{bf R}}&fg=000000$ or $latex {{bf C}}&fg=000000$, as in this previous blog post, but we will restrict attention here to the classical situation of real or complex arithmetic functions.) Experience has shown that a particularly tractable and relevant class of arithmetic functions for analytic number theory are the multiplicative functions, which are arithmetic functions $latex {f: {bf N} rightarrow {bf C}}&fg=000000$ with the additional property that

$latex displaystyle f(nm) = f(n) f(m) (1)&fg=000000$

whenever $latex {n,m in{bf N}}&fg=000000$ are coprime. (One also considers arithmetic functions that are not genuinely multiplicative, such as the logarithm function $latex {L(n) := log n}&fg=000000$ or the

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viXra Math Papers Publishing Site for Anybody

“arXiv” opposite is “viXra”.

The former “arXiv” is administered by Cornell University for Math paper publishing online. The traditional math journals would take 2 years to review and publish.

The Russian Mathematician G. Perelman was fed up of the long and bureaucratic review process, sent his proof of the 100-year-old unsolved “Poincaré Conjecture” to arXiv site. Later it was recognized to be correct, but Perelman refused to accept the Fields Medal and $1 million Clay Prize.

The new site “viXra” is open to  anybody in the world, while “arXiv” is still restricted to academia.

This young Singaporean published his new found Math Theorem on “viXra” site:

Prove that: if p is prime, for any integer k \geq 1

\boxed {(p - 1)^{p^k} \equiv -1 \mod {p^k}}

[By using the Binomial Theorem and Legendre’s Theorem.]

My Alternate Proof [Hint] : by using Graduate Advanced Algebra “Galois Finite Field Theory“:
Let q = p^k, where p prime and k >=1, it can be proved that GF(p^k) is the Field extension of GF(p).

Note: We say that p is the characteristic, k the dimension, of the Galois Field GF(p^k) of order (size) p^k.

Example: p = 3, k=2, 3^2=9
2^9 = 512 = -1 (mod 9)

Definition: (Without much frightening jargons, for a layman to understand): A Field is a number structure which allows {+ , *} and the respective opposite operations {-, ÷ }.

More intuitively, any Field numbers can be computed on a calculator with {+, -, ×, ÷} 4 basic operations.

It is a German term Körper , translated as Field (English), Corps (French), (Chinese / Japanese).

Examples of some standard Fields : Rational numbers (Q), Real numbers (R), Complex numbers (C).

Note 1: This diagram below explains what it means by Extension (or Splitting) Field:
Q is Rational Number Field (a, b in Q)
By extension (or splitting) we obtain new sub-Fields : eg.
1 +  \frac {3}{4}\sqrt {2}, \frac {1}{2} - 5 \sqrt {3}, ..., a+b\sqrt {n}

image

Note 2: Characteristic of GF(2), the Binary Field {0,1} is 2 because:
1+1 = 0 (1 add 2 times)
or 2 x (1) = 0

P.S. The ancient Chinese ‘magic’ game Chinese 9-Linked-Rings (九连环) is using the advanced Math Galois Field GF(2).
Baguenaudier Chinese Rings:

K = Field = GF(2)
p = 2 = characteristic of K
k = 9 = dimension of K-vector space

image

Music and Mathematics are Apolitical

This “Butterfly Lovers Violin Concerto” (梁山伯与祝英台: 梁祝 小提琴协奏曲 ) composed 50 years ago by 2 Chinese music students, now played so lovely by a Japanese lady violinist: 诹访内晶子 (Akiko Suwanai), who is the current user of the violin ”Dolphin”, one of the top 3 violins in the world made by Antonio Stradivarius

Only in the kingdom of Music (the other one is Mathematics) where human political hatred does not exist between countries due to past wars: Japan and China, Germany and the Allied Nations, … Just only yesterday China President Xi and Japan PM Abe both showed awkward “poker face” hand-shake at the APEC Beijing meeting; contrast to the 20th century’s greatest mathematician David Hilbert from Nazi Germany was welcome  in America to chair  the inauguration of the International Congress of Mathematics.


If more students love Math and Music, the world of tomorrow will be more peaceful.

Watch 諏訪內晶子 -《梁祝小提琴協奏曲》   Butterfly Lovers Violin Concerto. Notice her 1714 ‘magical’ violin is the treasure gift “Dolphin Stradivarius”, once owned and played by the virtuoso Jascha Heifetz (1901–1987).

Notes:
The legendary love story is the Chinese version of “Romeo & Juliet“:

http://en.m.wikipedia.org/wiki/Butterfly_Lovers’_Violin_Concerto

Interview with Akiko Suwanai

The history of the 2 composers (1958)