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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 07 Feb 2024 01:55:08 +0000A Centennial History of the Prime Number Theorem
https://resolver.caltech.edu/CaltechAUTHORS:20190913-091354267
Authors: {'items': [{'id': 'Apostol-T-M', 'name': {'family': 'Apostol', 'given': 'Tom M.'}}]}
Year: 2000
DOI: 10.1007/978-93-86279-02-6_1
Among the thousands of discoveries made by mathematicians over the centuries, some stand out as significant landmarks. One of these is the prime number theorem, which describes the asymptotic distribution of prime numbers. It can be stated in various equivalent forms, two of which are:
π(x) ∼ x/log x as x → ∞, (1) and p_n ∼ n log n as n → ∞.
(2)https://authors.library.caltech.edu/records/t9d0s-48077Computer Animated Mathematics Videotapes
https://resolver.caltech.edu/CaltechAUTHORS:20190828-102317036
Authors: {'items': [{'id': 'Apostol-T-M', 'name': {'family': 'Apostol', 'given': 'Tom M.'}}]}
Year: 2002
DOI: 10.1007/978-3-642-56240-2_1
Visualization — the representation of ideas, principles or problems by images — has always played an important role in both teaching and learning mathematics. Visual images make a much greater impact than printed or spoken words. People tend to forget words that they read or hear, but images are retained for a long time because they have emotional as well as intellectual appeal. This is especially true when the images are in motion and are accompanied by music and sound effects. The impact of well-crafted televised images on the human mind has been exploited by entertainers, advertisers, and politicians since the advent of television.https://authors.library.caltech.edu/records/vk7gq-afd12Euler Sums Revisited
https://resolver.caltech.edu/CaltechAUTHORS:20190909-133031057
Authors: {'items': [{'id': 'Apostol-T-M', 'name': {'family': 'Apostol', 'given': 'Tom M.'}}]}
Year: 2003
DOI: 10.1007/978-1-4615-0304-0_14
A large literature exists relating Riemann's zeta function
ζ(s)=∑k=1 ∞ 1/ks,R(s)>1,
and partial sums of the harmonic series,
h(n)=∑k=1 n 1/k.
Much of the research originated from two striking formulas discovered by Euler,
∑n=1 ∞ h(n)/n^2=2ζ(3)
(1)
,
∑n=1 ∞ h(n)/n^3=54ζ(4),
(2)
and a recursion formula, also due to Euler, which states that for integer a ≥2 we have
(a+1/2)ζ(2a)=∑k=1 a−1 ζ(2k)ζ(2a−2k).
(3)
For a = 2 and a =3 this gives ζ(4) =2/5ζ(2)^2 and ζ(6) =8/35ζ(2)^3. More generally, it shows that ζ(2) is a rational multiple of ζ(2) n. These results were rediscovered and extended by Ramanujan [11] and many others [1][5][6][8].https://authors.library.caltech.edu/records/2tdjp-4h423A Proof that Euler Missed: Evaluating ζ(2) the Easy Way
https://resolver.caltech.edu/CaltechAUTHORS:20191008-140350094
Authors: {'items': [{'id': 'Apostol-T-M', 'name': {'family': 'Apostol', 'given': 'Tom M.'}}]}
Year: 2004
DOI: 10.1007/978-1-4757-4217-6_51
R. Apéry [1] was the first to prove the irrationality of ζ(3)=∑ ∞ n=1 1/n^3.https://authors.library.caltech.edu/records/ed9ka-jqy34