#A08 INTEGERS 9 (2009), ON ASYMPTOTIC CONSTANTS RELATED TO PRODUCTS OF BERNOULLI NUMBERS AND FACTORIALS

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1 #A08 INTEGERS 9 009), ON ASYMPTOTIC CONSTANTS RELATED TO PRODUCTS OF BERNOULLI NUMBERS AND FACTORIALS Bernd C. Kellner Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, Göttingen, Germany bk@bernoulli.org Received: 9/4/07, Accepted: //09 Abstract We discuss the asymptotic expansions of certain products of Bernoulli numbers and factorials, e.g., B ν and kν)! νr as n for integers k and r 0. Our main interest is to determine exact expressions, in terms of known constants, for the asymptotic constants of these expansions and to show some relations among them.. Introduction Let B n be the nth Bernoulli number. These numbers are defined by z e z = z n B n, z < π n! n=0 where B n = 0 for odd n >. The Riemann zeta function ζs) is defined by ζs) = p s ), s C, Re s >. ) ν s = p By Euler s formula we have for even positive integers n that ζn) = πi) n B n. ) n! Products of Bernoulli numbers occur in certain contexts in number theory. For example, the Minkowski Siegel mass formula states that, for positive integers n with 8 n, Mn) = B k k k B ν 4ν, n = k,

2 INTEGERS: 9 009) 84 which describes the mass of the genus of even unimodular positive definite n n matrices for details see [, p. 5]). We introduce the following constants which we shall need further on. Lemma. There exist the constants C = C = C 3 = ζν) = , ν= ζν) = , ζν + ) = Proof. We have log + x) < x for real x > 0. Then log ζν) = log ζν) < ζν) ) = ) The last sum of 3) is well known and follows by rearranging in geometric series, since we have absolute convergence. We then obtain that π /6 < C < e 3/4, ζ3) < C 3 < C, and C = C C 3. To compute the infinite products above within a given precision, one can use the following arguments. A standard estimate for the partial sum of ζs) is given by ζs) N ν s < N s, s R, s >. s This follows by comparing the sum of ν s and the integral of x s in the interval N, ). Now, one can estimate the number N depending on s and the needed precision. However, we use a computer algebra system, that computes ζs) to a given precision with already accelerated built-in algorithms. Since ζs) monotonically as s, we next have to determine a finite product that suffices the precision. From above, we obtain ζs) < s + ), s s R, s >. 4) According to 3) and 4), we then get an estimate for the remainder of the infinite product by log ζν) < N +ε ν>n where we can take ε = 3/N ; the choice of ε follows by x + x log and 4).

3 INTEGERS: 9 009) 85 We give the following example where the constant C plays an important role; see Finch [8]. Let an) be the number of non-isomorphic abelian groups of order n. The constant C equals the average of the numbers an) by taking the limit. Thus, we have C = lim N N N an). By definition the constant C is connected with values of the Riemann zeta function on the positive real axis. Moreover, this constant is also connected with values of the Dedekind eta function ητ) = e πiτ/ on the upper imaginary axis. Lemma. The constant C is given by /C = p where the product runs over all primes. n= e πiντ ), τ C, Im τ > 0 p η i log p ) π Proof. By Lemma and the Euler product ) of ζs), we obtain C = p ν ) = p p p ν ) where we can change the order of the products because of absolute convergence. Rewriting p ν = e πiντ with τ = i log p /π yields the result. We used Mathematica [7] to compute all numerical values in this paper. The values were checked again by increasing the needed precision to 0 more digits.. Preliminaries We use the notation f g for real-valued functions when lim x fx)/gx) =. As usual, O ) denotes Landau s symbol. We write log f for logfx)). Definition 3. Define the linear function spaces Ω n = span {x ν, x ν log x}, n 0 0 ν n

4 INTEGERS: 9 009) 86 over R where f Ω n is a function f : R + R. Let Ω = n 0 Ω n. Define the linear map ψ : Ω R which gives the constant term of any f Ω. For the class of functions F x) = fx) + Ox δ ), f Ω n, n 0, δ > 0 5) define the linear operator [ ] : CR + ; R) Ω such that [F ] = f and [F ] Ω n. Then ψ[f ]) is defined to be the asymptotic constant of F. We shall examine functions h : N R which grow exponentially; in particular these functions are represented by certain products. Our problem is to find an asymptotic function h : R + R where h h. If F = log h satisfies 5), then we have [log h] Ω n for a suitable n and we identify [log h] = [log h] Ω n in that case. Lemma 4. Let f Ω n where fx) = α ν x ν + β ν x ν log x) ν=0 with coefficients α ν, β ν R. Let gx) = fλx) with a fixed λ R +. Then g Ω n and ψg) = ψf) + β 0 log λ. Proof. Since gx) = fλx) we obtain gx) = α ν λx) ν + β ν λx) ν log λ + log x)). ν=0 This shows that g Ω n. The constant terms are α 0 and β 0 log λ, and thus ψg) = ψf) + β 0 log λ. Definition 5. For a function f : R + R we introduce the notation fx) = ν f ν x) with functions f ν : R + R in case f has a divergent series expansion such that fx) = m f ν x) + θ m x)f m x), θ m x) 0, ), m N f, where N f is a suitable constant depending on f.

5 INTEGERS: 9 009) 87 Next we need some well-known facts which we state without proof see [0]). Proposition 6. Let H 0 = 0, H n = ν, n be the nth harmonic number. These numbers satisfy H n = γ+log n+on ), n, where γ = is Euler s constant. Proposition 7. Stirling s series) The Gamma function Γx) has the divergent series expansion log Γx + ) = logπ) + x + ) log x x + B ν νν ) x ν ), x > 0. ν Remark 8. When evaluating the divergent series given above, we have to choose a suitable index m such that ν B ν νν ) x ν ) = m B ν νν ) x ν ) + θ m x)r m x) and the remainder θ m x)r m x) is as small as possible. Since θ m x) 0, ) is not effectively computable in general, we have to use R m x) instead as an error bound. Schäfke and Finsterer [5], among others, showed that the so-called Lindelöf error bound L = for the estimate L θ m x) is best possible for positive real x. Proposition 9. If α R with 0 α <, then Γn + α) π ν α) = Γ α) Γ α) Euler s formula for the Gamma function states the following. n e ) n n α as n. Proposition 0. Euler) Let Γx) be the Gamma function. Then n ν Γ = n) n π). n Proposition. Glaisher [9], Kinkelin []) As n, ν ν A n nn+)+ e n 4, where A = is the Glaisher Kinkelin constant, which is given by log A = ζ ) = γ + logπ) ζ ) π.

6 INTEGERS: 9 009) 88 Numerous digits of the decimal expansion of the Glaisher Kinkelin constant A are recorded as sequence A07496 in OEIS [6]. 3. Products of Factorials In this section we consider products of factorials and determine their asymptotic expansions and constants. For these asymptotic constants we derive a divergent series representation as well as a closed formula. Theorem. Let k be a positive integer. Asymptotically, we have ) k nn+) kν)! F k A k π) k n n 4 πke k/ n) n e 3/ 4 + k + k as n with certain constants F k which satisfy log F k = γ k + B j ζj ) jj ) k j. j Moreover, the constants have the asymptotic behavior that lim F k =, k lim F k k = e γ/, k and F k F n γ/ as n k= with log F = γ + B j ζj ). jj ) j Theorem 3. If k is a positive integer, then log F k = k + ) log A + k k k log k + k k 4 logπ) ν ν ) k log Γ. k We will prove Theorem 3 later, since we shall need several preliminaries. Proof of Theorem. Let k be fixed. By Stirling s approximation, see Proposition 7, we have logkν)! = logπ) + kν + ) logkν) kν + fkν) 6) where we can write the remaining divergent sum as fkν) = kν + B j jj ) kν) j. j

7 INTEGERS: 9 009) 89 Define Sn) = + + n = nn + )/. By summation we obtain logkν)! = n logπk) + log n! ksn) + ksn) log k + k ν log ν + fkν). The term log n! is evaluated again by 6). Proposition provides that k ν log ν = k log A + ksn) log n + k log n k Sn) n ) + On δ ) with some δ > 0. Since lim n H n log n = γ, we asymptotically obtain for the remaining sum that lim n n ) fkν) k log n = γ k + B j ζj ) jj ) k j =: log F k. 7) j Here we have used the following arguments. We choose a fixed index m > for the remainder of the divergent sum. Then lim n B m θ m kν) mm ) kν) m = η B m ζm ) m mm ) k m 8) with some η m 0, ), since θ m kν) 0, ) for all ν. Thus, we can write 7) as an asymptotic series again. Collecting all terms, we finally get the asymptotic formula logkν)! = log F k + k log A + 4 logπ) + ksn) 3 ) + logkn) logπk) + k + log n ) + n k + ) log n + On δ ) k with some δ > 0. Note that the exact value of δ does not play a role here. Now, let k be an arbitrary positive integer. From 7) we deduce that log F k = The summation of 7) yields γ k + Ok 3 ) and k log F k = γ + Ok ). 9) log F k = γ H n + k= B j ζj ). 0) jj ) kj k= j

8 INTEGERS: 9 009) 90 Similar to 7) and 8), we can write again: n ) log F k γ log n lim n k= = γ + B j ζj ) =: log F. ) jj ) j The case k = of Theorem is related to the so-called Barnes G-function see []). Now we shall determine exact expressions for the constants F k. For k this is more complicated. Lemma 4. We have F = π) 4 e /A. Proof. Writing down the product of n! repeatedly in n + rows, one observes by counting in rows and columns that From Proposition 7 we have n + ) log n! = n + n! n+ = ν! ν ν. ) logπ) nn + ) + n + ) n + ) log n + + On ). Comparing the asymptotic constants of both sides of ) when n, we obtain π) e = F A π) 4 A where the right side follows by Theorem and Proposition. Proposition 5. Let k, l be integers with k. Define F k,l n) := kν l)! for 0 l < k. Then [log F k,l ] Ω and F k,0 n) F k,k n) = F,0 kn). Moreover F k,l n)/f k,l+ n) = k n n ν l ) k and [logf k,l /F k,l+ )] = [log F k,l ] [log F k,l+ ] Ω. for 0 l < k Proof. We deduce the proposed products from kν l)!/kν l + ))! = kν l and kν)!kν )! kν k ))! = k ν!. 3)

9 INTEGERS: 9 009) 9 Proposition 9 shows that [logf k,l /F k,l+ )] Ω. Since the operator [ ] is linear, it follows that [logf k,l /F k,l+ )] = [log F k,l log F k,l+ ] = [log F k,l ] [log F k,l+ ] Ω. 4) From Theorem we have [log F k,0 ] Ω. By induction on l and using 4) we derive that [log F k,l ] Ω for 0 < l < k. Lemma 6. Let k be an integer with k. Define the k k matrix M k := where all other entries are zero. Then det M k = k and the matrix inverse is given by M k = M k k with k k k 3 k k 3 k 3 M k = k ) 3 k ) k ) Proof. We have det M =. Let k 3. We recursively deduce by the Laplacian determinant expansion by minors on the first column that det M k = ) + det M k + ) +k det T k where the latter matrix T k is a lower triangular matrix having in its diagonal. Therefore det M k = det M k + ) +k ) k = k + = k by induction on k. Let I k be the k k identity matrix. The equation M k M k = k I k is easily verified by direct calculation, since M k has a simple form. Proof of Theorem 3. The case k = agrees with Lemma 4. For now, let k. We use the relations between the functions F k,l, resp. log F k,l, given in Proposition 5. Since [log F k,l ] Ω, we can work in Ω. The matrix M k defined in Lemma 6 mainly describes the relations given in 3) and 4). Furthermore we can reduce

10 INTEGERS: 9 009) 9 our equations to R by applying the linear map ψ, since we are only interested in the asymptotic constants. We obtain the linear system of equations M k x = b, x, b R k where x = ψ[log F k,0 ]),..., ψ[log F k,k ])) T and b = b,..., b k ) T with b l+ = ψ[logf k,l /F k,l+ )]) = logπ) log Γ l ) k for l = 0,..., k using Proposition 9. The last element b k is given by Theorem, Lemma 4, and Lemma 4: b k = ψ[logf,0 kn))]) = 4 logπ) + log F + log A + 5 log k = logπ) log A log k. By Lemma 6 we can solve the linear system directly with The first row yields On the other side, we have This provides x = k M k b. x = k b k + k k ν) b ν. k x = ψ[log F k,0 ]) = log F k + logπ) + k log A. 4 log F k = k + ) k log A + k 4 + k ) logπ) + 5 k log k + k k ν= ν k ν ) 5) log Γ k after some rearranging of terms. By Euler s formula, see Proposition 0, we have k ν ) log Γ = k k ) logπ) log k. 6) k k Finally, substituting 6) into 5) yields the result.

11 INTEGERS: 9 009) 93 Remark 7. Although the formula for F k has an elegant short form, one might also use 5) instead, since this formula omits the value Γ/k). Thus we easily obtain the value of F from 5) at once: F = π) e 4 /A 5. Corollary 8. Asymptotically, we have n ν ν e Γ n) γ A π) 4 A ) n / n as n with the constants e γ /A = and π) 4 /A = Proof. On the one hand, we have by 9) that n log F n = γ + On ). On the other hand, Theorem 3 provides that n log F n = n + ) log A + n n log n + 4 logπ) ν ν log Γ. n) Combining both formulas easily gives the result. Since we have derived exact expressions for the constants F k, we can improve the calculation of F. The divergent sum of F, given in Theorem, is not suitable to determine a value within a given precision, but we can use this sum in a modified way. Note that we cannot use the limit formula n ) log F = lim log F k γ n log n k= without a very extensive calculation, because the sequence γ n = H n log n converges too slowly. Moreover, the computation of F k involves the computation of the values Γν/k). This becomes more difficult for larger k. Proposition 9. Let m, n be positive integers. Assume that m > and the constants F k are given by exact expressions for k =,..., n. Define the computable values η k 0, ) implicitly by log F k = γ m k + j= B j ζj ) jj ) k j + η B m ζm ) k mm ) k m.

12 INTEGERS: 9 009) 94 Then log F = γ m + j= with θ n,m θ min n,m, θ max n,m) 0, ) where B j ζj ) jj ) + θ n,m B m ζm ) mm ) θ min n,m = ζm ) k= η k k m, θmax n,m = + ζm ) k= η k k m. The error bound for the remainder of the divergent sum of log F is given by n ) θn,m err = ζm ) k= k m B m ζm ). mm ) Proof. Let n and m > be fixed integers. The divergent sums for log F k and log F are given by Theorem. Since we require exact expressions for F k, we can compute the values η k for k =,..., n. We define and η m,k = η m,k = η k for k =,..., n η m,k = 0, η m,k = for k > n. We use 0) and ) to derive the bounds: θ min n,m = ζm ) k= η m,k k m < θ n,m < ζm ) k= η m,k n,m. = θmax km We obtain the suggested formulas for θn,m min and θn,m max by evaluating the sums with η m,k = 0, resp. η m,k =, for k > n. The error bound is given by the difference of the absolute values of the minimal and maximal remainder. Therefore θ err n,m = θ max n,m θ min n,m)r = ζm ) n k= k m ) R with R = B m ζm ) /mm ). Result 0. Exact expressions for F k : F = π) 4 e /A, F = π) e 4 /A 5, F 3 = π) e 36 /A 0 3 Γ 3 ) 3, F 4 = π) 7 3 e 48 /A 4 Γ 3 ) 4. We have computed the constants F k by their exact expression. Moreover, we have determined the index m of the smallest remainder of their asymptotic divergent series and the resulting error bound given by Theorem.

13 INTEGERS: 9 009) 95 Constant Value m Error bound F F F F F F The weak interval of F is given by Theorem. The second value is derived by Proposition 9 with parameters m = 7 and n = 7. Thus, exact expressions of F,..., F 7 are needed to compute F within the given precision. Constant Value / Interval m Error bound F.048,.049) F Products of Bernoulli Numbers Using results of the previous sections, we are now able to consider several products of Bernoulli numbers and to derive their asymptotic expansions and constants. Theorem. Asymptotically, we have B ν B n πe 3/ ) nn+) 6πn) n n 4 as n, B ν ν with the constants B n πe 3/ ) n 4n πe ) n / n 4 as n B = C F A π) 4 = C π) 5 4 e 4 /A, B = C F A /π) 4 = C 5 4 e 4 /A. Proof. By Euler s formula ) for ζν) and Lemma we obtain n B ν C ν)! π) ν C n π) nn+) ν)! as n. Theorem states for k = that ν)! F A π) 4 ) nn+) n 4πn) n n e 3/ 4 as n.

14 INTEGERS: 9 009) 96 The expression for F is given in Remark 7. Combining both asymptotic formulas above gives the first suggested formula. It remains to evaluate the following product: ν) = n n! π) ) n n n as n. e After some rearranging of terms we then obtain the second suggested formula. Remark. Milnor and Husemoller [4, pp ] give the following asymptotic formula without proof: where B ν B n! n+ F n + ) as n 7) n ) n F n) = πe 3/ 4 ) n 8πe 4 / n 4 8) n and B is a certain constant. This constant is related to the constant B. Proposition 3. The constant B is given by Proof. By Theorem we have B = 4 3 B = C e 4 / 5 4 A = B ν ν B Gn) as n 9) with Gn) = n πe 3/ ) n 4n πe We observe that 7) and 9) are equivalent so that We rewrite 8) in the suitable form F n + ) = Hence, we easily deduce that ) n / n 4. B F n + ) B Gn) as n. Gn)/F n + ) = n + )n +n+ ) n 4 4πe + 4 / πe 3/ n ) n n + 4 n e 4 n n + ) 4. e / 4 ) 4.

15 INTEGERS: 9 009) 97 It is well known that lim + x ) n = e x n n and lim n e xn + x ) n = e x. n Evaluating the asymptotic terms, we get B /B Gn)/F n + ) e 8 e 4 4 e 8 as n, which finally yields B = 4 3 B. Theorem 4. The Minkowski Siegel mass formula asymptotically states for positive integers n with 4 n that Mn) = B n n with B 3 = B. n B ν 4ν n ) n / ) n 4n B 3 n πe 3/ 4 as n πe Proof. Let n always be even. By Proposition 7 and ) we have n B n /n B n /n = ζn) n! πn ζn) n)! since ζn)/ζn) and ) n! log n n log n n)! We finally use Theorem and 9) to obtain Mn) = n B n /n B n /n B ν ν ) n 4n as n, πe n + ) log as n. ) n 4n B Gn) as n, πe which gives the result. Result 5. The constants B, B ν ν =,, 3) mainly depend on the constant C and the Glaisher Kinkelin constant A. Constant Expression Value A C B C π) 5 4 e 4 /A B C 5 4 e 4 /A B 3 C 7 4 e 4 /A B C e 4 / 5 4 A

16 INTEGERS: 9 009) Generalizations In this section we derive a generalization of Theorem. The results show the structure of the constants F k and the generalized constants F r,k, which we shall define later, in a wider context. For simplification we introduce the following definitions which arise from the Euler-Maclaurin summation formula. The sum of consecutive integer powers is given by the well-known formula n ν r = B r+n) B r+ r + ν=0 = r j=0 r n )B j+ r j j j +, r 0 where B m x) is the mth Bernoulli polynomial. Now, the Bernoulli number B = is responsible for omitting the last power n r in the summation above. Because we further need the summation up to n r, we change the sign of B in the sum as follows: S r n) = ν r = r j=0 This modification also coincides with r ) ) r j n j+ B r j j j +, r 0. ζ n) = ) n+ B n+ n + for nonnegative integers n. We define the extended sum S r n; f )) = r j=0 ) r ) r j n j+ fj + ) B r j, r 0 j j + where the symbol is replaced by the index j + in the sum. Note that S r is linear in the second parameter, i.e., Finally we define S r n; α + βf )) = αs r n) + βs r n; f )). D k x) = j Bj,k x j ) where Bm,k = B m mm )k m. Theorem 6. Let r be a nonnegative integer. Then ν νr A r Q r n) as n,

17 INTEGERS: 9 009) 99 where A r is the generalized Glaisher Kinkelin constant defined by Moreover, log Q r Ω r+ with log A r = ζ r) H r ζ r). log Q r n) = S r n) ζ r)) log n + S r n; H r H ). Proof. This formula and the constants easily follow from a more general formula for real r > given in [0, 9.8, p. 595] and after some rearranging of terms. Remark 7. The case r = 0 reduces to Stirling s approximation of n! with A 0 = π. The case r = gives the usual Glaisher Kinkelin constant A = A. The expression S r n; H r H ) does not depend on the definition of B, since the term with B is cancelled in the sum. Graham, Knuth, and Patashnik [0, 9.8, p. 595] notice that the constant ζ r) has been determined in a book of de Bruijn [7, 3.7] in 970. The theorem above has a long history. In 894 Alexeiewsky [3] gave the identity ν νr = exp ζ r, n + ) ζ r)) where ζ s, a) is the partial derivative of the Hurwitz zeta function with respect to the first variable. Between 903 and 93, Ramanujan recorded in his notebooks [5, Entry 7, pp ] the first part was published and edited by Berndt [5] in 985) an asymptotic expansion for real r > and an analytic expression for the constant C r = ζ r). However, Ramanujan only derived closed expressions for C 0 and C r r ) in terms of ζr + ); see 8) below. In 933 Bendersky [4] showed that there exist certain constants A r. Since 980, several others have investigate the asymptotic formula, including MacLeod [3], Choudhury [6], and Adamchik [, ]. Theorem 8. Let k, r be integers with k and r 0, then kν)! νr F r,k A r A k r+ P r,k n) Q r n) Qr+ n) k as n. The constants F r,k and functions P r,k satisfy that lim k F r,k = and log P r,k Ω r+ where log P r,k n) = S rn) logπk) + k S r+ n) logk/e) + B r+,k log n + r+ j= B j,k S r+ j n). The constants A r and functions Q r are defined as in Theorem 6.

18 INTEGERS: 9 009) 00 The determination of exact expressions for the constants F r,k seems to be a very complicated and extensive task in the case r > 0. The next theorem gives a partial result for k = and r 0. Theorem 9. Let r be a nonnegative integer, then log F r, = log A r log A r+ + S r ; B +, log A ). Case r = 0: Case r > 0: where α r,j = log F r, = + log A 0 log A. r+ log F r, = α r,0 + α r,j log A j j= B r+ rr+) r r ) Br j B j+ j j+) j+) j=0 ) Br+ j r+ δ r+,j r+ j, r j mod ), j = 0;, r j mod ), j = 0;, r j mod ), j > 0; 0, r j mod ), j > 0 and δ i,j is Kronecker s delta. Proof of Theorem 8. Let k and r be fixed. We extend the proof of Theorem. From 6) we have logkν)! = ) k logπk) + kν log + kν + ) log ν + D k ν). 0) e The summation yields ν r logkν)! = F n) + F n) + F 3 n) where F n) = S rn) logπk) + ks r+ n) logk/e), F n) = k F 3 n) = ν r+ log ν + ν r D k ν). ν r log ν,

19 INTEGERS: 9 009) 0 Theorem 6 provides F n) = k log A r+ + log Q r+ n)) + log A r + log Q r n)) + On δ ) with some δ > 0. Let R = r+. By definition we have x r D k x) = R B j,k x r+ j + Bj,k x j>r r+ j =: E x) + E x). j= Therewith we obtain that F 3 n) = R B j,k S r+ j n) + j= E ν). For the second sum above we consider two cases. We use similar arguments which we have applied to 7) and 8). If r is odd, then lim n E ν) = j>r Bj,k ζj r + )). ) Note that B r+,k = 0 in that case. If r is even, then we have to take care of the term ν. This gives lim n n ) E ν) B r+,k log n = γ B r+,k + Bj,k ζj r + )). ) j>r+ The right hand side of ), resp. ), defines the constant log F r,k. Finally we have to collect all results for F, F, and F 3. This gives the constants and the function P r,k. It remains to show that lim k log F r,k = 0. This follows by B j,k 0 as k. The following lemma gives a generalization of Equation ) in Lemma 4. After that we can give a proof of Theorem 9. Lemma 30. Let n, r be integers with n and r 0. Then n n! Srn) ν νr = n ν! νr ν Srν). 3)

20 INTEGERS: 9 009) 0 Proof. We regard the following enumeration scheme which can be easily extended to n rows and n columns: r r 3 r r r 3 r 3r 3r 3 3r The product of all elements, resp. non-framed elements, in the νth row equals n! νr, resp. ν! νr. The product of the framed elements in the νth column equals ν Srν ). Thus n n! Srn) = ν! νr ν Srν) νr. Proof of Theorem 9. Let r 0. We take the logarithm of 3) to obtain where F n) + F n) = F 3 n) + F 4 n) 4) F n) = S r n) log n!, F n) = F 3 n) = ν r log ν!, F 4 n) = ν r log ν, S r ν) log ν. Next we consider the asymptotic expansions F j of the functions F j j =,..., 4) when n. We further reduce the functions F j via the maps CR + ; R) [ ] Ω ψ R to the constant terms which are the asymptotic constants of [ F j ] in Ω. Consequently 4) turns into ψ[ F ]) + ψ[ F ]) = ψ[ F 3 ]) + ψ[ F 4 ]). 5) We know from Theorem 6 and Theorem 8 that ψ[ F ]) = log A r and ψ[ F 3 ]) = log F r, + log A r + log A r+. For F 4 we derive the expression ψ[ F 4 ]) = S r ; log A ), 6) since each term s j ν j in S r ν) produces the term s j log A j. It remains to evaluate F.

21 INTEGERS: 9 009) 03 According to 0) we have log n! = logπ) n + n + ) log n + D n) =: En) + D n). Thus F x) = S r x)ex) + S r x)d x). Since S r E Ω has no constant term, we deduce that ψ[ F ]) = ψ[s r D ]) = S r ; B +, ). The latter equation is derived as was6), except that we regard the constant terms of the product of the polynomial S r and the Laurent series D. From 5) we finally obtain log F r, = log A r log A r+ + S r ; B +, log A ). Now, we shall evaluate the expression above. For r = 0 we get log F 0, = + log A 0 log A, since S 0 ; B +, log A ) = B, log A = log A. For now, let r > 0. We may represent log F r, in terms of log A j as follows: The term α r,0 is given by r+ log F r, = α r,0 + α r,j log A j. α r,0 = S r ; B +, ) = r j=0 j= ) r ) r j Bj+, B r j j j + where the sum runs over even j, since B j+, = 0 for odd j. If r is odd, then the sum simplifies to the term B r+ /rr + ). Otherwise we derive for even r that α r,0 = r ) r Br j B j+ j j + ) j + ). j=0 It remains to determine the coefficients α r,j for r + j. Since xr x r+ S r x) is an odd, resp. even, polynomial for even, resp. odd, r > 0, this property

22 INTEGERS: 9 009) 04 transfers in a similar way to log A r log A r+ S r ; log A ), such that α r,j = 0 when r j. Otherwise we get ) ) r Br j ) r + Br+ j α r,j = δ r+,j = j j j r + δ r+,j 7) for r j, where the term log A r+ is represented by δ r+,j. Corollary 3. Let r be an odd positive integer, then r r! log F r, = ζr + ) r + )ζr + ) πi) r+ + ζr + j)ζj + ) r j= r = ) r r! B r+ rr + )! + B r+ j ζj + ) r + )ζr + ) r + j)! π) j π) r+. j= Proof. As a consequence of the functional equation of ζs) and its derivative, we have for even positive integers n that log A n = ζ n) = n! ζn + ) 8) πi) n where the left hand side of 8) follows by definition [5, p. 76]. Theorem 9 provides log F r, = B r+ rr + ) + α r,j log A j. Combining 7) and 8) gives the second equation above. By Euler s formula ) we finally derive the first equation. Remark 3. For the sake of completeness, we give an analogue of 8) for odd integers. From the logarithmic derivatives of Γs) and the functional equation of ζs), see [5, pp. 83, 76], it follows for even positive integers n, that where log A n = B n n H n ζ n) = B n )! γ + logπ)) + n n πi) n ζ n) ζ n) = r+ j= logν) ν n. ν= However, Mathematica is able to compute values of ζ for positive and negative argument values to any given precision.

23 INTEGERS: 9 009) 05 Result 33. Exact expressions for F r, in terms of A j : Constant Expression Value F 0, e A 0 A F, e 4 A F, e A 6 A F 3, e 70 A 4 A F 4, e A 30 A 3 3 A F 5, e 50 A A 4 A Exact expressions for F r, in terms of ζj + ): F, = exp F 3, = exp F 5, = exp 4 3ζ3) 8π ), ), 70 ζ3) 6π + 5ζ5) 6π ζ3) 48π + 5ζ5) 6π 4 05ζ7) 6π 6 For the first 5 constants F r, r = 0,..., 4) we find that max F r, < 0.05, 0 r 4 but, e.g., F 9, 37.6 and F 0, ). Acknowledgements. The author would like to thank Steven Finch for informing us about the formula of Milnor Husemoller and the problem of finding suitable constants; also for giving several references. The author is also grateful to the referee for several remarks and suggestions. References [] V. S. Adamchik, Polygamma functions of negative order, J. Comput. Appl. Math ), [] V. S. Adamchik, On the Barnes function, Proc. 00 Int. Symp. Symbolic and Algebraic Computation, Academic Press, 00, 5 0. [3] W. Alexeiewsky, Ueber eine Classe von Functionen, die der Gammafunction analog sind, Sitz.ber. Sächs. Akad. Wiss. Leipzig Math.-Nat.wiss. Kl ), [4] L. Bendersky, Sur la fonction gamma généralisée, Acta Math ), 63 3.

24 INTEGERS: 9 009) 06 [5] B. C. Berndt, Ramanujan s Notebooks, Part I, Springer-Verlag, 985. [6] B. K. Choudhury, The Riemann zeta-function and its derivatives, Proc. Royal Soc. London A ), [7] N. G. de Bruijn, Asymptotic Methods in Analysis, third edition 970. Reprinted by Dover, 98. [8] S. R. Finch, Abelian group enumeration constants, Mathematical Constants, Cambridge Univ. Press, 003, [9] J. W. L. Glaisher, On the product 3 3 n n, Messenger of Math ), [0] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, USA, 994. [] H. Kinkelin, Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechnung, J. Reine Angew. Math ), -58. [] M. Koecher, A. Krieg, Elliptische Funktionen und Modulformen, Springer-Verlag, 998. [3] R. A. MacLeod, Fractional part sums and divisor functions, J. Number Theory 4 98), [4] J. Milnor, D. Husemoller, Symmetric bilinear forms, Springer-Verlag, 973. [5] F. W. Schäfke, A. Finsterer, On Lindelöf s error bound for Stirling s series, J. Reine Angew. Math ), [6] N. J. A. Sloane, Online Encyclopedia of Integer Sequences OEIS), electronically published at: [7] Wolfram Research Inc., Mathematica, Wolfram Research Inc., Champaign, IL.