01. Simplest example phenomena

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1 (March, 20) 0. Simlest examle henomena Paul Garrett htt:// garrett/ There are three tyes of objects in lay: rimitive/rimordial (integers, rimes, lattice oints,...) zeta functions, L-functions modular/automorhic forms The rimitive objects are familiar, and accessible to everyone: Z = integers = {..., 3, 2,, 0,, 2, 3,...} rime numbers = {2, 3, 5, 7,, 3, 7,...} [0.] Questions about integers and rimes It is easy to formulate questions about integers and rimes, though not easy to distinguish those questions that have deeer meanings. More than 2000 years ago, contemoraries of Euclid knew that there are infinitely-many rime numbers. Quantifying the distribution of rimes is still essentially an oen question! Chebyshev (848-50), Riemann (857-8), Hadamard (896), and de la Vallée-Poussin (896) touched various asects of the basic Prime Number Theorem / x lim π(x) x log x = (where π(x) = number of rimes less than x) That is, in Landau s big-oh/little-oh notation, [] there is an asymtotic π(x) = x ( x ) log x + o log x Riemann s memoir suggested that the main term can be adjusted and a much more recise error term given, contingent on roerties of the zeta function (below). Very little rogress has been made in this direction! In 837 Dirichlet used L-functions (below) to rove that there are infinitely-many rimes in arithmetic rogressions a, a + m, a + 2m, a + 3m,..., whenever the obviously-necessary condition gcd(a, m) = holds. It is not known whether there are infinitely-many rimes of the form n 2 +. It is not known whether there are infinitely-many twin rimes, that is, rimes such that + 2 is also rime, although around 95 Viggo Brun roved that < while Euler had roven twin rimes all rimes = [] For two functions f, g, say f = o(g) as x x o when lim x xo f(x)/g(x) = 0. Say f = O(g) as x x o when lim su x xo f(x)/g(x) <.

2 Paul Garrett: 0. Simlest examle henomena (March, 20) More to the oint, we do not know connections of many such questions to anything else. [0.2] Zeta and L-functions: simlest examle The simlest zeta function is the (Euler-) Riemann zeta function ζ(s) = n s (comlex variable s) n= Convergence for Re(s) > follows immediately by comarison with the integral dx [ ] T x s = lim T (s ) x s = lim ( T s T s) The first non-trivial assertion about ζ(s) is that it has an Euler roduct exansion n n s = This follows from exanding the geometric series rimes s = + s + 2s + 3s +... s and using unique factorization of ositive integers into rimes. This factorization roduces an exression involving just rimes equated to an exression not overtly involving rimes. This hints at the relevance of the zeta function to rime numbers. The Euler roduct for ζ(s) is the entry to non-elementary study of rime numbers by analysis. Whether or not we care greatly about rime numbers, the connection between rimes and behavior of ζ(s) as a function of a comlex variable will be striking: see Riemann s Exlicit Formula a bit later. The first qualitative result about rimes was Euler s of about 750, using the Euler roduct exansion of ζ(s), roving = all rimes The roof is by calculus: first, comarison with x s dx for real s > roves that ζ(s) + as s +. On the other hand, from log( x) = x + x 2 /2 + x 3 /3 +..., log ζ(s) = log ( s ) = ( s + 2 2s + 3 3s +... ) The terms after the initial / s do not contribute to any blow-u as s +, as is seen by crudely estimating rimes by ositive integers: l ls ln ls dx l l 2 n 2 l 2 l 2 x ls = l 2 l ls l 2 l l < (for s > ) Thus, for some finite constant C, log ζ(s) C + s (for s > ) 2

3 Paul Garrett: 0. Simlest examle henomena (March, 20) Thus, the blow-u of log ζ(s) as s + forces the divergence [2] of /. Dirichlet s roof of the infinitude of rimes in arithmetic rogressions a, a + N, a + 2N,... is a refinement of Euler s argument, with a new ingredient, characters of abelian grous. A Dirichlet L-function mod N is L(s, χ) = n χ(n) n s where χ is a C-valued function defined modulo N, that is, χ(n + N) = χ(n) (for all n =, 2, 3,...) and χ is multilicative in the sense that χ(ab) = χ(a) χ(b) For fixed ositive integer n, Hurwitz zeta function is defined on ositive-definite symmetric real matrices Y by ζ n (Y, s) = (l Y l) s 0 l Z n In fact, in terms of its function, this turns out to be a degenerate Eisenstein series, so is really an automorhic form. [0.3] Evaluation of ζ(2), ζ(4),... About 750, the state of the art was that no one could understand the number = ζ(2) or other values of ζ(s). In that context, the following heuristic of Euler s was a sensation: imagine that sin x has a roduct exansion analogous to the roduct exansion of olynomials, given their zeros, ( x 2 ) sin x = x = x x 3 (πn) 2 n n= ζ(2) π = x x3 n2 π On the other hand, we know the ower series exansion Thus, equating coefficients, aarently sin x = x x3 3! + x5 5!... 3! Slightly messier maniulations yield all values ζ(2k). = ζ(2) π 2 Nowadays, many things are known about secial values of L-functions, but many mysteries remain. For examle, in 978 Aéry roved that ζ(3) is irrational, but very little more is known about bad values of L-functions, such as ζ(3). Altogether, desite more than 50 years of study, there are many unresolved questions about ζ(s), and about every other reasonable zeta and L-function! [2] For real s >, certainly s /, so the blow-u of the left-hand side as s + imlies divergence of the right. 3

4 Paul Garrett: 0. Simlest examle henomena (March, 20) [0.4] Modular forms Modular forms are also known as automorhic forms, and by many other similar names. They are hard to exlain or characterize, but have been crucial in many develoments in modern mathematics, esecially number theory. A few examles can be written down directly, but their significance is not clear from their exressions. The simlest modular forms are functions on the comlex uer half-lane H. Three of the simlest tyes are (holomorhic) Eisenstein series = E 2k (z) = (cz + d) 2k (z H, 2 < 2k 2Z) corime c,d Z,c 0 theta series = θ k (z) = e πiz(l l 2 k ) (z H, n Z) l Z k Im(z) s (waveform) Eisenstein series = E s (z) = cz + d 2s (z H, s C) corime c,d Z,c 0 Ramanujan s = (z) = e 2πiz ( e 2πinz ) 24 (z H) n Aart from being functions on the uer half-lane, it is not clear what these have in common. Theta series have immediate relevance to more tangible things, since they can be re-arranged to be generating functions counting the exressions of non-negative integers as sums of k squares. Secifically, let Then ν k (n) = number of (l,..., l k ) Z k such that n = l l 2 k θ k (z) = ν k (n) e πiz ν k(n) One might have observed that the k th theta series in this family is the k th ower of the first one: l Z k n=0 e πiz(l l 2 k ) = ( e πizl2) k An archetye for the relation of modular/automorhic forms to zeta functions and L-functions is the following connection between the simlest theta series θ(z) = l Z and the zeta function. Recall [3] eπizl2 Euler s integral for the gamma function Γ(s) = 0 y s y dy e y l Z (interolating factorials, since Γ(n) = (n )!) The relation of interest, known to Riemann and robably much earlier, is the integral reresentation of ζ(s) in terms of θ(z): 2π s/2 Γ( s 2 ) ζ(s) = y s/2 ( θ(iy) ) dy y This relation is essential in roof of basic roerties of ζ(s), such as analytic continuation, functional equation, and vertical growth estimates. This archetyical examle is one starting oint for alications of automorhic forms to zeta functions, and then to rime numbers. [0.5] The -function It is a little unfortunate that the symbol is over-used, but here the exlanation is that this arose historically as essentially the discriminant of Weierstraß cubic equation (u) 2 = 4 (u) 3 g 2 (u) g 3 (for u C) 0 [3] When these notes say recall it is not to imly dereliction on the art of the reader if the discussion is unfamiliar. 4

5 Paul Garrett: 0. Simlest examle henomena (March, 20) for Weierstraß ellitic function attached to the lattice Λ = Z + Z z deending on z H (u) = z λ Λ ( (z + λ) 2 ) λ 2 (for lattice Λ = Z + Z z in C) The symbols g 2, g 3 are traditional, and slightly mask the relation to the holomorhic Eisenstein series listed above: g 2 = g 2 (z) = 60 λ 4 = 60 2ζ(4) E 4(z) g 3 0 λ Λ = g 3 (z) = 40 0 λ Λ λ 3 = 40 2ζ(6) E 4(z) Then, u to choices of normalization, and not at all obviously, (z) really is the discriminant of the cubic, namely (z) = ( E4 (z) 3 E 6 (z) 2) 728 The roduct exansion of (z) is a non-trivial assertion, and needs a secial argument. The -function was studied for many decades before Ramanujan, but he found many, many unexected connections to secial functions and combinatorics, so his name is often associated to this examle. [0.6] Siegel-Weil formulas The very different aearance of theta series and Eisenstein series gives no hint that there is any relation between them. To give the simlest examle of such a relationshi, a slightly adjusted theta series is needed. Instead of sums-of-squares, a different quadratic form is needed. Some other ossibilities are suggested by rewriting the sum of k squares as l l 2 k = l l = l... l (viewing l as a column vector) Generalizing the identity matrix to a general (symmetric) k-by-k matrix Q, the family of functions includes the sums-of-squares as a secial case. q(l,..., l k ) = l Q l Because of some technical requirements we won t discuss just yet, for our examle, we take [4] Q = (unmarked entries are 0) [4] Elaborating a few details, although still not exlaining why: we need an 8-by-8 symmetric ositive-definite integer matrix Q with 2 s on the diagonal and with determinant. Admittedly, it is not obvious why we need this, or that it is ossible, but one should try to view these requirements as innocent variations on the sum-of-squares theme. The rationale for the secifics will be clear later. It s not obvious that there exists such a thing, or how to find it, but the examle given is one. It s not trivial to verify that it satisfies these conditions, either! 5

6 Paul Garrett: 0. Simlest examle henomena (March, 20) and instead of counting exressions of n as a sum of 8 squares, we count exressions n = l Q l = 8l 2 + 6l l 2 + 2l l 2 l 3 + 2l l 3 l 4 + 2l l 7 l 8 + 2l 8 (l j Z) Not as aealing or simle as the sum of eight squares would be. But, with this adatation, form the analogous theta series θ Q (z) = e πiz l Ql (with z H) l Z 8 Amazingly, this theta series is exactly an Eisenstein series: θ Q (z) = E 4 (z) This is crazy! This is the simlest Siegel-Weil relation. We will see that this relation has a more down-toearth corollary, giving another roof of π = ζ(4) = [0.7] Chaos? The above samle henomena merely touch the surface of many dee relationshis. Without knowing more, these examles are admittedly chaotic and of unclear imort. Subsequent discussion will assimilate them into a larger, coherent icture. 6