AND THE KUMMER CONGRUENCES

JORDAN SCHETTLER

Abstract. We use the language of the hyperreal to show how the well-known Kummer congruences between rational numbers involving Bernoulli numbers follow from an obvious congruence of infinite hyperintegers. Further, we make Euler’s intuitions on divergent series precise and show how the values of the Riemann zeta at negative agree with the values associated to divergent series. We do this by demonstrating that Ramanujan summation corresponds, in a certain sense, to taking a “standard part” of an infinite hyperreal .

Contents 1. Motivation 1 2. A Brief History of the Riemann Zeta Function 3 3. Ramanujan Summation 8 4. Hyperreal Numbers 9 References 14

1. Motivation For an even k > 2, the nonnormalized Eisenstein series of weight k is defined as X 1 G (τ) = k (mτ + n)k m,n∈Z (m,n)6=(0,0) for τ in the complex upper half-plane H. It is easy to check that

Gk(τ + 1) = Gk(τ)(1.1) k Gk(−1/τ) = τ Gk(τ).

In fact, Gk(τ) is modular of weight k for the full modular group Γ = SL2(Z), i.e., Gk(τ) is holomorphic on the extended upper half-plane H∗ = H ∪ Q ∪ {i∞} and 1−k k Gk(γz) = det(γ) (cτ + d) Gk(τ)

a b az+b for all γ = ( c d ) ∈ Γ where γz = cz+d . Note that the transformation property follows from 1 1 0 1 Eq.s 1.1 above since Γ = h( 0 1 ) , ( −1 0 )i. Since Γ acts transitively on the Q ∪ {i∞}, the condition of being holomorphic at these cusps is equivalent to being holomorphic at i∞, which amounts to having a Fourier expansion ∞ X n Gk(τ) = anq n=0 1 2 JORDAN SCHETTLER

2πiτ for all q = e in a neighborhood of q = 0. We can compute the value a0 of Gk(τ) at τ = i∞: X 1 X 1 a0 = lim Gk(it) = lim = = 2ζ(k) t→∞ t→∞ (mit + n)k nk m,n∈Z n∈Z (m,n)6=(0,0) n6=0 where ζ(s) is the Riemann zeta function (described below). As we will note, Euler proved that (2π)kB 2(2πi)k  B  2ζ(k) = (−1)k/2+1 k = − k k! (k − 1)! 2k

where Bk is a nonzero . Moreover, using the partial fraction decomposition of the cotangent, one can show that for each n ≥ 1 2(2πi)k a = σ (n) n (k − 1)! k−1

where each σk−1(n) is actually an integer given by a divisor sum: for any s ∈ C, we define X s σs(n) = d d|n with the sum extending over all positive integer divisors d of n. These observations lead to the normalized Eisenstein series of weight k ∞ (k − 1)! Bk X E (τ) = G (τ) = − + σ (n)qn. k 2(2πi)k k 2k k−1 n=1 At the beginning of Chapter 7 in Haruzo Hida’s book [Hid93], we find this family of nor- malized Eisenstein series used to motivate the occurrence and study of p-adic families of modular forms for a prime p. These p-adic families are roughly speaking, modular forms fk whose Fourier coefficients depend p-adic analytically on the weight k. In this concrete case of Eisenstein series, we can easily deduce p-adic continuity conditions for positive index coefficients from the following observation. If p is an odd prime and d is an integer not divisible by p, then for any integer m ≥ 1 dϕ(pm) ≡ 1 (mod pm)

by the Euler-Fermat theorem where ϕ is the Euler totient function. Thus if k1, k2 are integers m with k1 ≡ k2 (mod ϕ(p )), then (p) X X (p) σ (n) := dk1−1 ≡ dk2−1 = σ (n) (mod pm)(1.2) k1−1 k2−1 d|n,p-d d|n,p-d

− ordp(x) for all positive integers n. Let |x|p = p denote the normalized p-adic absolute value of a p-adic rational x where ordp denotes the p-adic order. Then the congruence property in Eq. 1.2 can be restated in the equivalent form |σ(p) (n) − σ(p) (n)| < |k − k | k1−1 k2−1 p 1 2 p

whenever k1 ≡ k2 (mod p − 1) and k1 6= k2. Note that ∞ (p) Bk X (p) E (τ) := E (τ) − pk−1E (pτ) = − (1 − pk−1) + σ (n)qn. k k k 2k k−1 n=1 NONSTANDARD ANALYSIS AND THE KUMMER CONGRUENCES 3

(p) The constant term in the Fourier series of Ek also satisfies a p-adic continuity condition. Theorem 1 (Kummer Congruences). Suppose p is an odd prime. Then B B k1 k1−1 k2 k2−1 (1.3) − (1 − p ) + (1 − p ) < |k1 − k2|p 2k1 2k2 p

whenever k1 ≡ k2 6≡ 0 (mod p − 1) and k1 6= k2. In his book [Was96], Lawrence Washington proves these congruences as an easy corollary to the existence of p-adic L-functions attached to Dirichlet characters. One can also find elementary proofs, such as the one found in the book [IR90] by Ireland and Rosen, which use an interesting intermediate congruence due to Voronoi. We do not wish to use either of those approaches. Rather, we want to regard B − k (1 − pk−1) = ζ(1 − k)(1 − pk−1) 2k ∞ ∞ X 1 X 1 “ = ” − n1−k (pn)1−k n=1 n=1 X “ = ” dk−1 d|0,p-d (p) =: σk−1(0). The first equality follows from the functional equation (see 2.6 below) of ζ(s), but we must make sense of the equalities in quotation marks because they equate expressions involving divergent series. There should be a simple extension of the ideas above that will allows us to show |σ(p) (0) − σ(p) (0)| < |k − k | under similar conditions with the same reasoning. k1−1 k2−1 p 1 2 p (p) The idea is to treat σk−1(0) as a truly infinite integer, a hyperinteger in fact; we then want a natural way to associate a finite rational number to these infinite integers which agrees with the values coming from the meromorphic continuation of ζ(s), and we want to do this in a way such that the Kummer congruences are shadows of congruences for infinite integers.

2. A Brief History of the Riemann Zeta Function In 1735, solved the Basel problem by discovering the famous identity 1 1 1 1 π2 + + + + ··· = . 12 22 32 42 6 He reasoned that this follows from comparing the Taylor series of sin(x) with the famous sine product formula and comparing x2-terms: ∞ 1 1 sin(x) Y   x 2 1 − x2 + x4 − · · · = = 1 − 6 120 x nπ n=1 The product formula was later made precise by Weierstraß. More generally, Euler computed (using only the of infinite series) 1 1 1 1 (2π)2kB (2.1) + + + + ··· = (−1)k+1 2k 12k 22k 32k 42k 2(2k)! 4 JORDAN SCHETTLER

for all positive integers k where the Bernoulli numbers Bn are defined for integers n ≥ 0 by ∞ t X tn = B . et − 1 n n! n=0

It is immediate from this definition that Bn ∈ Q for all n ≥ 0, and we can easily compute 1 1 that, e.g., B0 = 1, B1 = − 2 , and B2 = 6 . Note that t t et + 1 − B t = · et − 1 1 2 et − 1

is an even function of t, so Bn = 0 for all odd n > 1. Also, Eq. 2.1 implies that the of even indexed Bernoulli numbers B2n for n ≥ 1 alternates in sign. Euler further “computed” the values of certain divergent series which also involved the Bernoulli numbers: for integers n ≥ 0 B (2.2) 1k + 2k + 3k + 4k + ··· “ = ”(−1)k k+1 . k + 1

He did this via Abel summation as follows. Let Va denote the space of (an) of P∞ n real numbers such that n=1 anx has a radius of convergence at least 1 and represents a function f(x) such that limt→1− f(x) exists. Then Va contains the space W of all sequences P∞ (an) such that n=1 an converges, and there is an R-linear map ∞ X n A: Va → R:(an) 7→ lim anx . x→1− n=1 P∞ Moreover, the restriction A|W associates to (an) the value of the convergent series n=1 an. Euler considered the function 1 1 2 e−t − e−2t + e−3t − · · · = = − et + 1 et − 1 e2t − 1 ∞ X tn−1 = B (1 − 2n) , n n! n=1 so that for all positive integers n A((1k, −2k, 3k, −4k,...)) = lim (1ke−t − 2ke−2t + 3ke−3t − · · · ) t→0+ dk 1 = (−1)k lim t→0+ dtk et + 1 B = (−1)k k+1 (1 − 2k+1). k + 1 We thus we would like to conclude (1k + 2k + 3k + 4k + ··· )(1 − 2k+1) = 1k + 2k + 3k + 4k + · · · − 2(2k + 4k + 6k + 8k + ··· ) B = 1k − 2k + 3k − 4k + ··· = (−1)k k+1 (1 − 2k+1). k + 1 NONSTANDARD ANALYSIS AND THE KUMMER CONGRUENCES 5

In this way, Euler knew that the values of convergent series P n−k for even integers k > 1 were related to the values of the divergent series P n−(1−k). Note that we cannot apply the Abel summation map A to (1k, 2k, 3k, 4k,...) directly since the dk 1 lim 1ke−t + 2ke−2t + 3ke−3t + 4ke−t + ···  = lim (−1)k t→0+ t→0+ dtk et − 1 does not exist as a finite number. For further information see [Har92]. Euler also noted that for s > 1 ∞ X 1 Y (2.3) = (1 − p−s)−1 ns n=1 p prime by the fundamental theorem of arithmetic. Bernhard Riemann took this as his starting point for his paper “On the Number of Primes Less than a Given Magnitude”. His idea was to regard ∞ X 1 ζ(s) = ns n=1 as a function of a complex variable s and the use complex/Fourier analysis to derive formulas and asymptotics for the prime counting function. The sum represents an analytic function in the half plane <(s) > 1. Riemann showed how ζ(s) can be meromorphically continued to the entire complex plane with the only pole being a simple pole at s = 1 having residue 1. Meromorphic continuations are unique when they exist, so, e.g., the values ζ(0), ζ(−1), ζ(−2), ... are uniquely determined and thus represent “appropriate” values of the associ- ated divergent series. We now review Riemann’s method of meromorphic continuation and computation of special values as found in [Edw01], for example. For s > −1 we define Z ∞ s! := e−xxs dx. 0 Integration by parts shows that n! = 1 · 2 · 3 ··· (n − 1)n for nonnegative integers n. For s > 1 ∞ Z ∞ X xs−1 dx (s − 1)!ζ(s) = e−x · ns−1 n 0 n=1 ∞ Z ∞ X = e−nxxs−1 dx 0 n=1 Z ∞ xs dx = x · , 0 e − 1 x so Z s Z ∞ s (−x) dx iπs −iπs x dx x · = (e − e ) x · C e − 1 x 0 e − 1 x = 2i sin(πs)(s − 1)!ζ(s) where C is the contour seen below. 6 JORDAN SCHETTLER

Now s! has a meromorphic continuation to C: 1 · 2 ··· (n − 1)n s! := lim (n + 1)s n→∞ (s + 1)(s + 2) ··· (s + n) There are poles of s! at s = −1, −2,... but no zeros. There is also a functional identity π sin(πs)(s − 1)! = , (−s)! whence (−s)! Z (−x)s dx (2.4) ζ(s) = x · 2πi C e − 1 x is meromorphic on all of C. Note that the poles of (−s)! for integers s > 1 must cancel with zeros from the integral since in those cases the defining series is convergent. We compute (directly!) for k = 0, 1, 2,... k! Z x (−x)−k dx ζ(−k) = x · · 2πi |x|=ε e − 1 x x ∞ k! X Bm = Res xm−k−2 (−1)k m! x=0 m=0 k! B B = · k+1 = (−1)k k+1 . (−1)k (k + 1)! k + 1 Thus ζ(0) = −1/2 and for k = 1, 2, ··· ζ(−2k) = 0 (trivial zeros) B (2.5) ζ(1 − 2k) = − 2k , 2k so these agree with Euler’s values associated to divergent series. In particular: B 1 1 + 2 + 3 + ··· “ = ”ζ(−1) = − 2 = − 2 12 NONSTANDARD ANALYSIS AND THE KUMMER CONGRUENCES 7

There is also a functional equation which Riemann derived by considering the boundary ∂R (seen below) of a region R ⊂ C containing the singularities of the integrand in the meromorphic continuation expression 2.4.

We can actually apply Cauchy’s theorem here to get ∞ Z (−x)s dx 2πi X −(−x)s−1 0 = · = − ζ(s) − 2πi Res , ex − 1 x (−s)! x=2πin ex − 1 ∂R n=−∞ n6=0 so ∞ X ζ(s) = −(−s)! (−(−2πin)s−1 − (2πin)s−1)(2.6) n=−1 ∞ X = (−s)!(2π)s−1((−i)s−1 + is−1) ns−1 n=1 = (−s)!(2π)s−12 sin(sπ/2)ζ(1 − s). This is the famous functional equation which can be rewritten in the more symmetric form ξ(s) = ξ(1 − s) where ξ(s) := s(1 − s)π−s/2(s/2 − 1)!ζ(s) is a “completed” zeta function. We can now recover the computations of Euler: for an integer k ≥ 1 we have from Eq.s 2.5 and 2.6 B − 2k = ζ(1 − 2k) = (2k − 1)!(2π)−2k2 sin((1 − 2k)π/2)ζ(2k) 2k (2k − 1)!2 = (−1)k ζ(2k). (2π)2k 8 JORDAN SCHETTLER

Note that the functional equation tells us nothing about the values of ζ(2k + 1) for integers k ≥ 1 since we have both B2k+1/(2k + 1) = 0 and sin((2k)π/2) = 0.

3. Ramanujan Summation In February of 1913, Srinivasa Ramanujan wrote to Cambridge mathematician G. H. Hardy and said...

“If I tell you 1 1+2+3+4+5+6+ ··· = − , 12 you will at once point out to me the lunatic asylum as my goal.”

We know from above that this is the correct value of ζ(−1) that one obtains from analytic continuation. Ramanujan did not know any complex analysis, however, nor did he use Abel summation to derive the value. Instead, he worked with the partial sums of the divergent series, viewing them as approximations to integrals via the trapezoidal rule. The remain- der terms (which involve Bernoulli numbers) can be computed up to an additive constant. Ramanujan wrote x Z x ∞ X 1 X B2n f(n) = C + f(t) dt + f(x) + f (2n−1)(x) 2 (2n)! n=1 0 n=1 and said “The constant [C] of a series has some mysterious connection with the given infinite series and it is like the centre of gravity of a body. Mysterious because we may substitute it for the divergent series.” There is an alternate method of achieving meromorphic continuation using Bernoulli poly- nomials which will simultaneously illustrate Ramanujan summation in the case f(x) = −s (x + 1) . Given B0(x) = 1, recursively define Bn(x) for n ≥ 1 by d Z 1 Bn(x) = nBn−1(x) and Bn(x) dx = 0 dx 0 2 It is clear that, e.g., B1(x) = x − 1/2 and B2(x) = x − x + 1/6. These polynomials allow us to give an equivalent definition of the Bernoulli numbers Bn = Bn(0). From the boundary conditions, we also see that Bn = Bn(1) for n > 1. Let bxc = x − {x} denote the largest integer which is less than or equal to x. We find for s 6= 1 N X 1  1 1   2 2   3 3   (N − 1) (N − 1) N = − + − + − + ··· + − + ns 1s 2s 2s 3s 3s 4s (N − 1)s N s N s n=1 N−1 1 X Z n+1 ns 1 Z N bxc = + dx = + s dx N s−1 xs+1 N s−1 xs+1 n=1 n 1 1 1 s Z N {x} = · s−1 + − s s+1 dx. 1 − s N s − 1 1 x 1 Note that if <(s) > 1, then N s−1 → 0 as N → ∞, so in this case s Z ∞ {x} ζ(s) = − s s+1 dx, s − 1 1 x NONSTANDARD ANALYSIS AND THE KUMMER CONGRUENCES 9 but the expression on the right hand side represents a meromorphic function in the half- plane <(s) > 0. Since meromorphic continuation is unique, we can compute the values of the Riemann zeta function in this larger half-plane using this formula. We want to continue to extend the half-plane of convergence of expressions for ζ(s), but we also wish to keep the information from the partial sums. This is where our first Bernoulli polynomial B1 makes an appearance. N X 1 1 1 s Z N 1/2 + ({x} − 1/2) = · + − s dx ns 1 − s N s−1 s − 1 xs+1 n=1 1 Z N 1 1−s 1 −s s 1 B1({x}) = N + N + − − s s+1 dx 1 − s 2 s − 1 2 1 x

As above, we get an expression for ζ(s), but now it remains valid in the half-plane <(s) > −1: Z ∞ s 1 B1({x}) ζ(s) = − − s s+1 dx s − 1 2 1 x

We can iterate this process using the recursive definition of the Bernoulli polynomials: N X 1 1 B1 B2 s B1 B2 (3.1) = N 1−s − N −s − s N −s−1 + + + s ns 1 − s 1! 2! s − 1 1! 2! n=1 Z N s(s + 1) B2({x}) − s+1 dx 2! 1 x r 1 X Bm = N 1−s − s(s + 1) ··· (s + m − 2) N 1−s−m 1 − s m! m=1 r s X Bm + + s(s + 1) ··· (s + m − 2) s − 1 m! m=1 Z N s(s + 1) ··· (s + r − 1) Br({x}) − s+1 dx. r! 1 x This gives convergence in the half-plane <(s) > −r if we ignore the terms with N, i.e., those terms which vanish as N → ∞ for s > 1.

4. Hyperreal Numbers We have seen above that one way of treating divergent series is to regard the terms as coefficients in a power series and assigning a value to the divergent series by taking a limit of the function so defined. This is not completely satisfactory, however, since such a limit will not always exist (as we have seen) in cases we care about. Moreover, it is not clear independent of Riemann’s calculations that the values obtained by Euler through Abel summation would/should agree with those from meromorphic continuation of ζ(s). P∞ A divergent series such as n=1 n should represent an infinite number, i.e., greater than any , because the partial sums tend to +∞. The reciprocal of this infinite number should then be a positive infinitesimal, i.e., less than any positive real number but greater 10 JORDAN SCHETTLER

than zero. In general, we will associate a “” to any sequence of real numbers such that sequences which diverge to ±∞ represent infinite numbers and sequences that approach zero represent infinitesimals. We want to be able to perform algebraic operations and order with inequalities on these hyperreal numbers just as we would with the real numbers. Moreover, the hyperreal numbers should contain a copy of the real numbers. One way to construct what we have described is by taking equivalence classes of sequences in such a way that the algebraic operations are well-defined and there are no zero divisors. We will say two sequences (aN ), (bN ) represent the same hyperreal number when the set of indexes N ∈ N such that aN = bN belongs to some distinguished collection C of subsets of N. Clearly, we want C to contain all complements of finite sets since two sequences which are identical at all but finitely many indexes should represent the same hyperreal number. It also seems reasonable to require a superset of a set in C to also belong to C since, for example, a sequence which approaches zero will still approach zero if we change any number its nonzero components to zero. Definition 2. A filter F of N is a collection of subsets of N satisfying the following properties: (1) If A ∈ F and A ⊆ B ⊆ N, then B ∈ F. (2) If A, B ∈ F, then A ∩ B ∈ F. (3) We have ∅ ∈/ F. We call a filter F an ultrafilter if it satisfies the following additional condition: (4) If A ⊆ N and A/∈ F, then N\A ∈ F.

Given an ultrafilter F of N, we will say two sequences (aN ) and (bN ) are equivalent if and only if

{N ∈ N : aN = bN } ∈ F, ∗ and we will denote equivalence classes by (aN ). Doing this will ensure that the algebraic operations are well-defined and that there are no zero divisors (ensured by the ultrafilter condition (4) above), but if F contains any finite set, then the equivalence classes of se- quences will just give us back the real numbers R. Filters which contain finite sets are called trivial. However, there are filters F which do not contain any finite sets. For example, the Fr´echet filter consisting precisely of the complements of finite sets is such a filter. Moreover, Zorn’s lemma implies that any filter is contained in an ultrafilter, so there exists a nontrivial ultrafilter U obtained by extending the Fr´echet filter. With this choice of U fixed, we can ∗ ∗ define the hyperreal numbers R as the set of equivalence classes (aN ) of sequences of real numbers (aN ). ∗ ∗ ∗ ∗ Theorem 3. The binary operations +, · on R given by (aN ) + (bN ) = (aN + bN ) and ∗ ∗ ∗ ∗ (aN )· (bN ) = (aN bN ) are well-defined. In fact, R is a field with respect to these operations and the natural map ∗ ∗ R → R: α 7→ (α, α, α, . . .) an an injective ring homomorphism. Given any real valued function f(x) of a real variable x, there is a natural extension to ∗ ∗ ∗ R given by f( (aN )) = (f(aN )). For example, the absolute value of a hyerreal number ∗ ∗ ∗ α = (α1, α2,...) is the hyperreal number |α| = (|αN |). Define the notation (aN ) ≤ (bN ) to ∗ mean {N ∈ N : aN ≤ bN } ∈ U. This gives a total ordering on R which is compatible with the field operations. We give ∗R the induced order topology. We always view R ⊂ ∗R via NONSTANDARD ANALYSIS AND THE KUMMER CONGRUENCES 11

the above injection, and the subspace topology on R is the familiar one. An element ε ∈ ∗R is called an infinitesimal if 0 < |ε| < |x| for all x ∈ R. An element of α ∈ ∗R is called finite if |α| < |x| for some x ∈ R. The set of finite hyperreals forms a ring F. For every α ∈ F, there is a unique real number a such that α − a is infinitesimal or zero; we call such an a the standard part of α and write st(α) = a. We get a surjective ring homomorphism st: F → R where ker(st) is the unique maximal ideal of F and consists precisely of the infinitesimals ∗ ∗ ∗ and zero. We should note that d( (aN ), (bN )) = st( (|aN − bN |)) does not define a metric on ∗R since, for example, d(x + ε, x) = 0 while x 6= x + ε for any nonzero infinitesimal ε. For any subset S ⊆ R, we take ∗S ⊆ ∗R to be the subset consisting of all equivalence ∗ ∗ classes (aN ) such that {N ∈ N : aN ∈ S} ∈ U. If S is a subring of R, then S is a subring of ∗R. We thus get the set of hypernatural numbers ∗N and rings of hyperintegers ∗Z and hyperrationals ∗Q. For s ∈ C, define N X s Ss(N) := n n=1 Then the function ∗ Z(s) = (S−s(N)) gives a well-defined map from R to ∗R such that st(Z(s)) = ζ(s) for all s > 1. On the other hand, how can we recover the values of ζ(s) for s < 1? Suppose that s = −k is a negative integer, and use Eq. 3.1 with r = k + 1 to get the familiar formula k+1 1 X Bm S (N) = N k+1 − (−1)(m−1)k(k − 1) ··· (k − m + 2) N k+1−m k k + 1 m! m=1 k+1 k X Bm + + (−1)m−1k(k − 1) ··· (k − m + 2) k + 1 m! m=1 (−1)k+1 (−1)k+1 = (−N) + 1 − (−1) k + 1 Bk+1 k + 1 Bk+1 (N) + (k + 1)N k (1) + (k + 1) · 1k =Bk+1 + 1 − Bk+1 k + 1 k + 1 (N + 1) B =Bk+1 − k+1 k + 1 k + 1

where we have used the fact (which follows from an easy inductive argument) that k+1 X k + 1 (x) = B xk+1−m. Bk+1 m m m=0 Then (N + 1) B Z(−k) = ∗(S (N)) = Bk+1 ∞ − k+1 ∈ [N ] k k + 1 k + 1 Q ∞ 12 JORDAN SCHETTLER

∗ ∗ where N∞ = (N) ∈ N is an infinite hyperinteger. Note that N∞ is transcendental over the real numbers R, so we have a well-defined R-linear map Z 0 ∼ F : R[N∞] = R[x] → R: f(N∞) ↔ f(x) 7→ f(x) dx −1 By definition Z 0   Z 1 Bk+1(x + 1) Bk+1 1 Bk+1 F (Z(−k)) = − dx = Bk+1(x) dx − −1 k + 1 k + 1 k + 1 0 k + 1 B = − k+1 = ζ(−k). k + 1 Fix a prime p ≥ 5. Obviously,

Z(−k1) ≡ Z(−k2) (mod p)

whenever k1 ≡ k2 (mod p−1). This is a congruence of infinite hyperintegers. In other words, Z(−k1) − Z(−k2) = pM for some infinite hyperinteger M. If, additionally, p − 1 - k1 + 1, then the Kummer congruences (Eq. 1.3) would imply that

F (Z(−k1)) ≡ F (Z(−k2)) (mod p) which is, in fact, a congruence of p-integers, i.e., rational numbers having p-adic absolute value less than or equal to 1 or, equivalently, rational numbers whose denominators when written in lowest terms are prime to p. More generally, consider ∗ −s Zp(s) = (S−s(pN) − p S−s(N)). −s Then st(Zp(s)) = (1 − p )ζ(s) for s > 1, while for negative integers s = −k we have  (pN + 1) − pk (N + 1) B  F (Z (−k)) = F Bk+1 ∞ Bk+1 ∞ − (1 − pk) k+1 p k + 1 k + 1 (−1)k+1 1k+1 + 2k+1 + ··· + (p − 1)k+1 = · + (1 − pk)ζ(−k). k + 1 p There is again an obvious congruence m Zp(−k1) ≡ Zp(−k2) (mod p ) m whenever k1 ≡ k2 (mod ϕ(p )). If, additionally, p−1 - k1 +1, then the Kummer congruences imply

k1 k2 m (1 − p )ζ(−k1) ≡ (1 − p )ζ(−k2) (mod p ), m but this congruence should NOT follow from Zp(−k1) ≡ Zp(−k2) (mod p ). Why? Let us take m = 1 from now on for convenience. Then for any integer k ≥ 1 p−1 X  0 if p − 1 k (4.1) nk ≡ - (mod p). −1 if p − 1|k n=1 Thus for each N pN p−1 X X  0 if p − 1 k nk = N nk ≡ - (mod p), −N if p − 1|k n=1 n=1 p-n NONSTANDARD ANALYSIS AND THE KUMMER CONGRUENCES 13

i.e.,  0 if p − 1 - k Zp(−k) ≡ (mod p) −N∞ if p − 1|k so the congruence Zp(−k1) ≡ Zp(−k2) (mod p) would not require the condition k1 ≡ k2 (mod p − 1) only that both or neither of the positive integers k1, k2 are zero modulo p − 1. However, we can “remove” the factor of pN∞. We are lead to define a modified Zp as follows:   pN p ˜ ∗  1 X −s n − n Zp(s − p) − Zp(s − 1) Zp(s) =  n  = . N p pN  n=1  ∞ p-n As before, we get an obvious congruence ˜ ˜ Zp(−k1) ≡ Zp(−k2) (mod p)(4.2)

whenever there is a congruence of finite positive integers k1 ≡ k2 (mod p − 1). Now we use an idea presented by Wells Johnson in [Joh75]. If p − 1 - k + 1 for a positive integer k, then p−1 X ω(n)k+1 = 0, n=1

where ω(n) is the unique root of unity in the p-adic integers Zp such that ω(n) ≡ n (mod n), whence p−1 p−1 k+1 1 X 1 X  ω(n) − n (4.3) 0 = ω(n)k+1 = n + p k + 1 k + 1 p n=1 n=1 p−1 k+1 p−1 j 1 X 1 X k + 1 X ω(n) − n = nk+1 + pj nk+1−j k + 1 k + 1 j p n=1 j=1 n=1 k+1   p−1  j k+2(p) − Bk+2 1 X k + 1 X ω(n) − n = B + pj nk+1−j (k + 1)(k + 2) k + 1 j p j=1 n=1 k+2 1 X k + 2 = B pj (k + 1)(k + 2) j k+2−j j=1 k+1 p−1 j 1 X k + 1 X ω(n) − n + pj nk+1−j k + 1 j p j=1 n=1

p−1 p Bk+1 X n − n ≡ p + p nk (mod p2) k + 1 p n=1 j where we have used the fact that ordp(p /j!) ≥ 2 for j ≥ 2. This is a congruence of p-integers which implies the congruence of hyperrationals B Z˜ (−k) ≡ − k+1 = ζ(−k) (mod p)(4.4) p k + 1 whenever p − 1 - k + 1. Of course, we did not need the hyperreals to deduce the Kummer congruences as Eq. 4.3 clearly suffices. However, the machinery of the hyperreals allows us 14 JORDAN SCHETTLER

to satisfactorily view the Kummer congruences as a finite shadow of congruences deduced only from the Euler-Fermat theorem applied to infinite numbers.

References [Edw01] H. M. Edwards, Riemann’s zeta function, Dover Publications Inc., Mineola, NY, 2001, Reprint of the 1974 original [Academic Press, New York; MR0466039 (57 #5922)]. [Har92] G. H. Hardy, Divergent series, Editions´ Jacques Gabay, Sceaux, 1992, With a preface by J. E. Littlewood and a note by L. S. Bosanquet, Reprint of the revised (1963) edition. [Hid93] Haruzo Hida, Elementary theory of L-functions and Eisenstein series, London Mathematical Soci- ety Student Texts, vol. 26, Cambridge University Press, Cambridge, 1993. [IR90] Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, second ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. [Joh75] Wells Johnson, p-adic proofs of congruences for the Bernoulli numbers, J. Number Theory 7 (1975), 251–265. [Was96] Lawrence C. Washington, Introduction to cyclotomic fields, second ed., Springer, 1996.