{"id":153,"date":"2008-11-18T21:00:41","date_gmt":"2008-11-18T20:00:41","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/2008\/11\/18\/the-harmonic-series-and-surrounding-folklore\/"},"modified":"2024-02-20T13:40:57","modified_gmt":"2024-02-20T11:40:57","slug":"the-harmonic-series-and-surrounding-folklore","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2008\/11\/18\/the-harmonic-series-and-surrounding-folklore\/","title":{"rendered":"The harmonic series and surrounding folklore"},"content":{"rendered":"

Speaking of divergent series<\/a>, I’d like to mention one of my favorites bits of informal folklore: this starts with a slogan<\/p>\n

The only “truly” divergent series is the harmonic series
\n$latex H=1+\\frac{1}{2}+\\cdots +\\frac{1}{n}+\\cdots,$<\/p><\/blockquote>\n

which, of course, deserves plenty of quotes. But the idea is quite sound: it is that, given about any other series with real or complex terms (which does not involve this one in a hidden manner, of course, e.g., as an “irreducible summand”), such as<\/p>\n

$latex \\sum_{m\\geq 1}{(-1)^{m+1}\\frac{m^2}{4m^2-1}},$<\/p>\n

it is quite often possible, by some kind of trick or feint, to assign a “value” (i.e., a real or complex number) to the series in a way which is reasonable and useful<\/em> for certain purposes. [A famous example is the (apparently) ultra-divergent series<\/p>\n

$latex \\sum_{m\\geq 0}{(-1)^m m!}=1-1+2-6+24-120+720-\\cdots,$<\/p>\n

which was considered by Wallis and Euler: its “value” is<\/p>\n

$latex e\\int_0^1{e^{-1\/x}x^{-1}dx}=e\\int_1^{+\\infty}{y^{-1}e^{-y}dy}=0.5963473623231\\ldots$<\/p>\n

(see this article<\/a> for an explanation of some of Euler’s ideas to do this “computation”).]<\/p>\n

One setting in which the philosophy above has been refined to a precise (conjectural) statement is the theory of L<\/em>-functions (over number fields). Indeed, observe that (formally) we have<\/p>\n

$latex H=\\zeta(1),$<\/p>\n

where \u256c\u00c2(s)<\/em> is the Riemann zeta function, which is only defined properly for the real part of s<\/em> larger than one by the series<\/p>\n

$latex \\zeta(s)=\\sum_{n\\geq 1}{\\frac{1}{n^s}}.$<\/p>\n

Then, after translating the basic insight about the harmonic series from Dirichlet series to L<\/em>-functions, one gets the following folklore conjecture:<\/p>\n

If a reasonable (say, automorphic, or “motivic”) L<\/em>-function L(s) over a number field k<\/em> has a pole at s=1<\/em> (when normalized so that the functional equation relates values at s<\/em> and 1-s<\/em>), then L(s)<\/em> is divisible by the Riemann zeta function<\/em>, in the sense that
\n$latex L_1(s)=\\frac{L(s)}{\\zeta(s)}$
\ndoes not<\/em> acquire poles at any of the complex zeros of the Riemann zeta function.<\/p><\/blockquote>\n

This is quite a rich problem: it contains a famous conjecture of Artin (the Dedekind zeta function of a number field should be divisible by the Riemann zeta function), and applied to the Rankin-Selberg convolution, it suggests the existence of the symmetric-square L<\/em>-functions of modular forms — indeed, the first crucial result towards the “automorphic” existence of the latter (due to Gelbart and Jacquet<\/a>), was the proof of this divisibility property by Shimura).<\/p>\n

Now for the most amazing thing concerning this folklore property (at least to me): it seems that it also works modulo primes! Let me explain this (an explanation which I heard from Jean-Pierre Serre): another renowned formula of Euler for the values of the zeta function at even integers can be rephrased, after using the functional equation<\/p>\n

$latex \\pi^{-s\/2}\\Gamma(s\/2)\\zeta(s)=\\pi^{-(1-s)\/2}\\Gamma((1-s)\/2)\\zeta(1-s),$<\/p>\n

as the formula<\/p>\n

$latex \\zeta(1-2n)=1+2^{2n-1}+3^{2n-1}+\\cdots + k^{2n-1}+\\cdots =-\\frac{1}{2n}B_{2n},$<\/p>\n

where the Bernoulli numbers<\/a> B2n<\/sub><\/em> are rational numbers defined by the power series expansion<\/p>\n

$latex 1-\\frac{1}{2}x+\\sum_{n\\geq 1}{B_{2n}\\frac{x^n}{n!}}=\\frac{x}{e^x-1}$<\/p>\n

(and, of course, the middle expression for \u256c\u00c2(1-2n)<\/em> is purely formal: it is one more example of a divergent series that can be given a convincing value).<\/p>\n

The idea now is that the last expression (-1\/2n B2n<\/sub><\/em>) can be reduced modulo a prime p<\/em>, provided p<\/em> does not divide 2n<\/em> and does not divide the denominator of the Bernoulli number. Now, lo and behold, the primes dividing the denominators of Bernoulli numbers are known: they are exactly the primes such that<\/p>\n

$latex p-1\\mid 2n$<\/p>\n

or equivalently such that<\/p>\n

$latex k^{2n-1}\\equiv k^{-1}\\text{ mod } p$<\/p>\n

for all integers k<\/em> coprime with p<\/em> (those for which the inverse modulo p<\/em> makes sense…). So the Bernoulli number can not be reduced modulo primes exactly when (either p<\/em> divides 2n<\/em> or) formally we have<\/p>\n

$latex -\\frac{1}{2n}B_{2n}\\equiv 1+2^{2n-1}+3^{2n-1}+\\cdots + k^{2n-1}+\\cdots\\equiv 1+2^{-1}+\\cdots +k^{-1}+\\cdots,$<\/p>\n

the divergent harmonic series again! (One must omit the terms with k<\/em> divisible by p<\/em> of course, but since this is purely formal, why not?).<\/p>\n

I have no idea if there is a good explanation for this coincidence, but it is remarkably beautiful, and it certainly gives a convincing argument for the fact that the numerators<\/em> of Bernoulli numbers are much more mysterious: so are the zeros of the Riemann zeta function, in comparison with its pole…)<\/p>\n","protected":false},"excerpt":{"rendered":"

Speaking of divergent series, I’d like to mention one of my favorites bits of informal folklore: this starts with a slogan The only “truly” divergent series is the harmonic series $latex H=1+\\frac{1}{2}+\\cdots +\\frac{1}{n}+\\cdots,$ which, of course, deserves plenty of quotes. But the idea is quite sound: it is that, given about any other series with … Continue reading The harmonic series and surrounding folklore<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-153","post","type-post","status-publish","format-standard","hentry","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/153","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=153"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/153\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=153"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=153"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=153"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}