11
votes
2answers
450 views

History of the analytic class number formula

The (general) analytic class number formula gives a value for the residue of the Dedekind zeta function of a number field at the point $s=1$ (or, as I prefer, the leading Taylor coefficient at $s=0$). ...
11
votes
3answers
2k views

Why are smooth numbers called “smooth”?

"Adleman refers to integers which factor completely into small primes as “smooth” numbers." (ME Hellman, JM Reyneri. Advances in Cryptology, 1983: citation link.) Does anyone know what is the ...
22
votes
2answers
1k views

Was Vinogradov's 1937 proof of the three-prime theorem effective?

Was Vinogradov's first proof of the three-prime theorem effective? Reasons for my question: Vinogradov presented his proof in 1937 in a monograph; the English translation by K.F. Roth and A. ...
6
votes
2answers
976 views

Ramanujan's tau function

Why was Ramanujan interested in the his tau function before the advent of modular forms? The machinery of modular forms used by Mordel to solve the multiplicative property seems out of context until I ...
0
votes
3answers
553 views

Definition of Prime Numbers [duplicate]

The first time I heard of prime numbers, they were defined as natural numbers $n$ that can only be divided by 1 and themselves without remainder; later, when prime factorization was introduced, I ...
8
votes
1answer
261 views

History question: Roth's theorem on approximating algebraic numbers…before Roth

Roth's theorem has two universal quantifies, over irrational algebraic numbers $\alpha$ and over real $\epsilon>0$. Of course the theorem asserts in each instance that the inequality ...
17
votes
1answer
480 views

What motivated Rademacher's contour along the Ford circles?

Apologies if this question isn't suitable for MathOverflow; I posted it on MSE here but it didn't get a response and it felt like it was on the cusp of being suitable for here. After Ramanujan and ...
11
votes
2answers
648 views

Le Haut Commissariat qui surveille rigoureusement l'alignement de ses Grandes Pyramides

Yesterday I came across the following one-paragraph summary of the history of the Law of Quadratic Reciprocity in Roger Godement's Analyse mathématique, IV, p.313 (perhaps the only treatise on ...
23
votes
0answers
738 views

Does a proof of Selberg's 3.2 inequality exist?

A well-known inequality of Montgomery and Vaughan (generalizing a result of Hilbert) states that $$ \left |\sum_{r \neq s} \frac{w_{r} \overline{w_{s}} }{\lambda_r - \lambda_s} \right| \leq \pi ...
21
votes
2answers
989 views

The Erdős-Turán conjecture or the Erdős' conjecture?

This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics: Conjecture: If $A\subset \mathbb{N}$ ...
6
votes
1answer
221 views

Did Smith correctly state the mass formula?

Did Smith correctly state the mass formula? H.J.S. "normal form" Smith was the first, in 1867, to state the mass formula for integral quadratic forms in a genus of 4 or more variables. This was ...
27
votes
3answers
2k views

What is the difference between an automorphic form and a modular form?

This is more of a question about terminology than about math. The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly which generalization it ...
11
votes
1answer
940 views

At what point does number theory stop playing with finite rings?

Basic results in number theory, like the Chinese remainder theorem, the Euclidean algorithm and Euler's theorem, are really about finite structures, namely the rings $\mathbb{Z}/n\mathbb{Z}$ for ...
10
votes
1answer
1k views

Did Gauss know Dirichlet's class number formula in 1801?

Let $h_d$ be the number of $SL_{2}(\mathbb{Z})$ classes of primitive binary quadratic forms of discriminant $d$. It's natural to impose the hypothesis that $d$ is not at square, as we do below. In ...
5
votes
3answers
1k views

First known proof of $\sqrt 2$ is irrational with prime factorization?

Do any of you happen to know the history of the standard prime factorization proof of $\sqrt 2$ is irrational? I know this theorem was known to Aristotle, and that the Fundamental Theorem of ...
31
votes
3answers
1k views

Definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel

(This question was posted on math.stackexchange a week ago at http://math.stackexchange.com/questions/187315/definitive-source-about-dirichlet-finally-proving-the-unit-theorem-in-the-sistinbut and ...
13
votes
2answers
704 views

Did Hermite really prove “Hermite's Theorem” on number field discriminants?

Hermite's theorem, as it is typically called, is that there are only finitely many number fields of bounded (equivalently, fixed) discriminant. The usual proof (see Neukirch's Algebraic Number Theory ...
3
votes
2answers
477 views

Who discovered the asymptotic formula for the number of partitions of n into distinct parts?

Who was the first to develop the asymptotic formulae for the distinct parts version of $p(n)?$
5
votes
1answer
256 views

Who proved that the plane partition generating function is valid?

I know Major Macmahon conjectured the formula $$ \prod_{m=1}^\infty \frac{1}{(1-q^m)^m}=1 + \sum_{n=1}^\infty PL(n)q^n$$ but who was the first to prove it?
14
votes
5answers
972 views

What is the “ray” in ray class group?

I have never seen any algebraic number theory book discuss the origin of the term "ray class group." Does anyone know where the word "ray" comes from in this context? I always thought it might be a ...
3
votes
2answers
1k views

Heuristics for the Hodge Conjecture

W. V. D. Hodge is famous for his Hodge conjecture, one of the Millennium prize problems. Hodge might have had some rough heuristics or ideas that led him to the formulation of the conjecture. I am ...
6
votes
3answers
1k views

Sum of the sum-of-divisors function

I was looking at the abstract of a paper [1] which claims that [2] and [3] prove $$ \sum_{n\le x}\sigma(n)-\frac{\pi^2}{12}x^2=\Omega(x\log\log x). $$ But I cannot find the above—or indeed, ...
7
votes
3answers
2k views

Riemann zeta at even integers

I am talking about this in a course I am teaching, and hence am wondering: what are the various derivations of the values of Riemann zeta function at even integers? There are two incredibly cool ...
11
votes
5answers
2k views

sums of rational squares

It is a well-known fact that if an integer is a sum of two rational squares then it is a sum of two integer squares. For example, Cohen vol. 1 page 314 prop. 5.4.9. Cohen gives a short proof that ...
31
votes
1answer
1k views

On a remark of Tait on FLT for the exponent 3

This is one of those recreational questions that aren't really about research. I found a curious remark in an old volume of American Mathematical Monthly (1922) which I'll quote below: In the ...
7
votes
1answer
383 views

Two implicit references in Serre's *Groupes de Galois : le cas abélien*

In his exposé at the Galois bicentenary conference, Serre makes two references which are not quite explicit. The first reference occurs (at 22:30 in the video) when he is talking about Dedekind's ...
9
votes
3answers
974 views

The first complete proof of the Kronecker-Weber theorem

While the Kronecker-Weber theorem —that every finite abelian extension of $\mathbb Q$ is contained in a cyclotomic field— is always attributed to, well, Leopold Kronecker and Heinrich Martin Weber, ...
6
votes
1answer
2k views

How did Birch and Swinnerton Dyer arrive at their conjecture?

I suspect that they knew that the $L-$function is defined only for $Re(s) \gt 3/2$. Did they attempt to evaluate the $L-$function at $s=1$ by plugging $s=1$ in the infinite product $\prod_p ...
5
votes
1answer
486 views

Kummer and Fermat's Equation

In Report on the Theory of Numbers, H.J.S. Smith writes: "The impossibility of solving [Fermat's] equation has been demonstrated by M. Kummer, first, for all values of $\lambda$ not included among ...
5
votes
5answers
2k views

Exponential sums for beginner.

What are the good books, online lecture notes or starting material on exponentials sums with applications in number theory for a beginner, apart from N. M. Korobov's book? The book or notes should ...
11
votes
4answers
8k views

The Ramanujan Problems.

I originally thought of asking this question at the Mathematics Stackexchange, but then I decided that I'd have a better chance of a good discussion here. In the Wikipedia page on Ramanujan, there is ...
19
votes
1answer
1k views

Fibonacci, compositions, history

There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets): a) compositions with parts from {1,2} (e.g., 2+2 = 2+1+1 = ...
23
votes
4answers
2k views

Overview of the interplay of Harmonic Analysis and Number Theory

I'm kind of disappointed that the question here was never sharpened. The Laplacian $\Delta$ on the upper half-plane is $-y^{2}(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}))$. Suppose $D$ ...
11
votes
2answers
703 views

Historical Articles about zeta functions of curves

Are there any historical articles about the origins of zeta functions of curves over global fields (undoubtedly starting with $\mathbb{Q}$)? In particular who (and when did this happen) first have ...
14
votes
6answers
2k views

A non-technical account of the Birch—Swinnerton-Dyer Conjecture

I was wondering whether anyone knows of any good non-technical or even popular expositions of the Birch—Swinnerton-Dyer conjecture, for someone with minimal background in elliptic curves. I was ...
15
votes
1answer
1k views

Primes represented by two-variable quadratic polynomials

I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its ...
13
votes
9answers
2k views

New proofs to major theorems leading to new insights and results?

I am wondering, historically, when has a new proof of an old theorem been particularly fruitful. A few examples I have in mind (all number theoretic) are: First example is classical... which is ...
2
votes
1answer
977 views

$230.$ (April, 1915) Proposed by E. B. Escott, Ann Arbor, Michigan

Just browsing some old stuff in my office for other thing I found the following: $230.$ (April, 1915) Proposed by E. B. Escott, Ann Arbor, Michigan. Find three numbers such that their sum, the sum ...
2
votes
1answer
423 views

Whence the k-tuple conjecture?

What is the source of the $k$-tuple conjecture, that every integer tuple $(k_1,\ldots,k_n)$ either contains all members of a congruence class mod a prime or has infinitely many primes amongst ...
3
votes
2answers
1k views

What is the shortest proof of the existence of a prime between $p$ and $p^2$ ? other examples? [closed]

1) It is well known that between a prime $p$ and $p^2$ always exist a prime, but what is the shortest proof of that (by elementary methods or not)? (One can say that we can have it as a collorary of ...
12
votes
2answers
3k views

Did Andre Weil predict that the Riemann Hypothesis would be settled by prime number theory rather than by analysis? If so, what are a reference and/or a quotation?

Did Andre Weil predict that the Riemann Hypothesis would be settled by prime number theory rather than by analysis? If so, what are a reference and/or a quotation?
15
votes
2answers
1k views

History of Irrationality results

The Greeks knew that numbers of the form $\sqrt{n}$ for nonsquare integers $n$ are not rational. Much later, Lambert (1768) proved that the values of $e^x$ and $\tan x$ are irrational for nonzero ...
30
votes
1answer
2k views

First correct proof of FLT for exponent 3?

It is well known that Euler gave the first proof of FLT ($x^n + y^n = z^n$ has no nontrivial integral solutions for $n > 2$) for exponent $n=3$, but that his proof had gaps (which are not as easily ...
8
votes
3answers
671 views

English or French translation of Gauss' “Summatio Quarumdam Serierum Singularium”

I'm interested in looking at the details of Gauss' method of determining the sign of the Gauss sum in his "Summatio Quarumdam Serierum Singularium", and I was wondering if anyone knew if there was an ...
44
votes
3answers
5k views

who fixed the topology on ideles?

I am teaching a course leading up to Tate's thesis and I told the students last week, when defining ideles, that the first topology that was put on the ideles was not so good (e.g., it was not ...
17
votes
6answers
1k views

A gamma function identity

In some of my previous work on mean values of Dirichlet L-functions, I came upon the following identity for the Gamma function: \begin{equation} \frac{\Gamma(a) \Gamma(1-a-b)}{\Gamma(1-b)} + ...
34
votes
3answers
2k views

What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory?

This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields ...
7
votes
7answers
2k views

Meaning of Kronecker's comment to Lindemann

At the Mactutor history page, it is said that Kronecker remarked to Lindemann: "What good your beautiful proof on [the transcendence of] π? Why investigate such problems, given that irrational ...
2
votes
2answers
497 views

Fermat for polynomials, as used in the AKS (Agrawal-Kayal-Saxena) algorithm

The basis for the deterministic polynomial-time algorithm for primality of Agrawal, Kayal and Saxena is (the degree one version of) the following generalization of Fermat's theorem. Theorem ...
11
votes
5answers
1k views

Christening Fermat's Little Theorem

I am writing an article on Fermat's work in number theory and feel uncomfortable everytime I have to write "Fermat's Little Theorem": it's clumsy and belittles the fundamental character of Fermat's ...