A201995 -id:A201995 - cp's OEIS frontend

A073002 Decimal expansion of -zeta'(2) (the first derivative of the zeta function at 2).

9, 3, 7, 5, 4, 8, 2, 5, 4, 3, 1, 5, 8, 4, 3, 7, 5, 3, 7, 0, 2, 5, 7, 4, 0, 9, 4, 5, 6, 7, 8, 6, 4, 9, 7, 7, 8, 9, 7, 8, 6, 0, 2, 8, 8, 6, 1, 4, 8, 2, 9, 9, 2, 5, 8, 8, 5, 4, 3, 3, 4, 8, 0, 3, 6, 0, 4, 4, 3, 8, 1, 1, 3, 1, 2, 7, 0, 7, 5, 2, 2, 7, 9, 3, 6, 8, 9, 4, 1, 5, 1, 4, 1, 1, 5, 1, 5, 1, 7, 4, 9, 3, 1, 1, 3

Offset: 0

Robert G. Wilson v, Aug 03 2002

Successive derivatives of the zeta function evaluated at x=2 round to (-1)^n * n!, for the n-th derivative, and converge with increasing n. For example, in Mathematica, Derivative[5][Zeta][2] = -120.000824333. A direct formula for the n-th derivative of Zeta at x=2 is: (-1)^n*Sum_{k>=1} log(k)^n/k^2. See also A201994 and A201995. The values of successive derivatives of Zeta(x) as x->1 are given by A252898, and are also related to the factorials. - Richard R. Forberg, Dec 30 2014

			Zeta'(2) = -0.93754825431584375370257409456786497789786028861482...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.18, p. 157.
C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 359.

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Huylebrouck, Generalizing Wallis' formula, American Mathematical Monthly, to appear, 2015.
Simon Plouffe, Zeta(1,2) the derivative of Zeta function at 2.
J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.

Cf. A201994 (2nd derivative), A201995 (3rd derivative), A252898.

Cf. A244115, A261506.

Zeta(1,2); evalf(%, 120); # R. J. Mathar, Oct 10 2011

(* first do *) Needs["NumericalMath`NLimit`"], (* then *) RealDigits[ N[ ND[ Zeta[z], z, 2, WorkingPrecision -> 200, Scale -> 10^-20, Terms -> 20], 111]][[1]] (* Eric W. Weisstein, May 20 2004 *)
(* from version 6 on *) RealDigits[-Zeta'[2], 10, 105] // First (* or *) RealDigits[-Pi^2/6*(EulerGamma - 12*Log[Glaisher] + Log[2*Pi]), 10, 105] // First (* Jean-François Alcover, Apr 11 2013 *)

-zeta'(2) \\ Charles R Greathouse IV, Mar 28 2012

Sum_{n >= 1} log(n) / n^2. - N. J. A. Sloane, Feb 19 2011

Pi^2(gamma + log(2Pi) - 12 log(A))/6, where A is the Glaisher-Kinkelin constant. - Charles R Greathouse IV, May 06 2013

Definition corrected by N. J. A. Sloane, Feb 19 2011

A201994 Decimal expansion of zeta''(2), the second derivative of the Riemann zeta function at 2.

Original entry on oeis.org

1, 9, 8, 9, 2, 8, 0, 2, 3, 4, 2, 9, 8, 9, 0, 1, 0, 2, 3, 4, 2, 0, 8, 5, 8, 6, 8, 7, 4, 2, 1, 5, 1, 6, 3, 8, 1, 4, 9, 4, 4, 6, 0, 7, 7, 0, 7, 4, 2, 5, 0, 2, 7, 0, 9, 0, 7, 8, 1, 5, 3, 9, 9, 2, 9, 2, 3, 0, 2, 5, 9, 4, 9, 3, 6, 3, 2, 3, 5, 4, 0, 2, 1, 1, 2, 6, 6, 8, 8, 8, 3, 4, 9, 0, 3, 5, 8

Offset: 1

R. J. Mathar, Dec 07 2011

			Equals 1.9892802342989010234208586874215163814944...

B. K. Choudhury, The Riemann zeta-function and its derivatives, Proc. R. Soc. Lond A 445 (1995) 477-499, Table 2 and (19).
Index entries for constants related to zeta

Cf. A073002, A201995, A340442 (at 3), A340443 (at 4).

Maple
```
evalf(Zeta(2,2),100) ;
```

RealDigits[ Zeta''[2], 10, 97] // First (* Jean-François Alcover, Feb 20 2013 *)

PARI
```
zeta''(2) \\ Michel Marcus, Jan 07 2021
```

zeta''(2) = Sum_{k>=1} (log(k)/k)^2.

A061502 a(n) = Sum_{k<=n} tau(k)^2, where tau = number of divisors function A000005.

Original entry on oeis.org

1, 5, 9, 18, 22, 38, 42, 58, 67, 83, 87, 123, 127, 143, 159, 184, 188, 224, 228, 264, 280, 296, 300, 364, 373, 389, 405, 441, 445, 509, 513, 549, 565, 581, 597, 678, 682, 698, 714, 778, 782, 846, 850, 886, 922, 938, 942, 1042, 1051, 1087

Offset: 1

N. J. A. Sloane, Jun 14 2001

nonn

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter II, Problem 56.

N. J. A. Sloane, Table of n, a(n) for n = 1..1024
Adrian Dudek, On the Success of Mishandling Euclid's Lemma, arXiv:1602.03555 [math.HO], 2016. See B(n) p. 2.
Chaohua Jia and Ayyadurai Sankaranarayanan, The mean square of the divisor function, Acta Arithmetica 164 (2014), 181-208.
Michaela Cully-Hugill and Timothy Trudgian, Two explicit divisor sums, arXiv:1911.07369 [math.NT], Nov 19 2019
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)
Florian Luca and László Tóth, The r-th moment of the divisor function: an elementary approach, Journal of Integer Sequences 20 (2017), Article 17.7.4, 8 pp.
Adolf Piltz, Über das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, 1881.
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (3).
D. Suryanarayana and R. Rama Chandra Rao, On an Asymptotic Formula of Ramanujan, Mathematica Scandinavica, 32, 258-264, 1973.
B. M. Wilson, Proofs of some formulas enunciated by Ramanujan, Proc. London Math. Soc. (2) 21 (1922) 235-255.

Cf. A000005, A035116, A061503, A318755.
Cf. A092742 (A), A245074 (B), A319090 (C), A319091 (D).
Cf. A057434, A072379, A074789.

[&+[NumberOfDivisors(k^2)*Floor(n/k): k in [1..n]]: n in [1..60]]; // Vincenzo Librandi, Sep 10 2016

Table[Sum[DivisorSigma[0, k^2]*Floor[n/k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 30 2018 *)
Accumulate[Table[DivisorSigma[0, n]^2, {n, 1, 50}]] (* Vaclav Kotesovec, Sep 10 2018 *)

for (n=1, 1024, write("b061502.txt", n, " ", sum(k=1, n, numdiv(k)^2)) ) \\ Harry J. Smith, Jul 23 2009

vector(60, n, sum(k=1, n, numdiv(k)^2)) \\ Michel Marcus, Jul 23 2015

first(n)=my(v=vector(n),s); forfactored(k=1,n, v[k[1]] = s += numdiv(k)^2); v; \\ Charles R Greathouse IV, Nov 28 2018

a(n) = Sum_{k=1..n} tau(k^2)*floor(n/k).
Asymptotic to A*n*log(n)^3 + B*n*log(n)^2 + C*n*log(n) + D*n + O(n^(1/2+eps)) where A = 1/Pi^2 and B = (12*gamma-3)/Pi^2 - 36*zeta'(2)/Pi^4. [corrected by Vaclav Kotesovec, Aug 30 2018]
C = 36*gamma^2/Pi^2 - (288*z1/Pi^4 + 24/Pi^2)*gamma + (864*z1^2/Pi^6 + 72*z1/Pi^4 - 72/Pi^4*z2 + 6/Pi^2) - 24*g1/Pi^2 and D = 24*gamma^3/Pi^2 - (432*z1 /Pi^4+ 36/Pi^2)*gamma^2 + (3456*z1^2/Pi^6 + 288*(z1-z2)/Pi^4 + 24/Pi^2 - 72*g1/Pi^2)*gamma + g1*(288*z1/Pi^4 + 24/Pi^2)-10368*z1^3/Pi^8 - 864*z1^2/Pi^6 + 1728*z2*z1/Pi^6 + 72*(z2-z1)/Pi^4- 48*z3/Pi^4 + (12*g2-6)/Pi^2, where gamma is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and g1, g2 are the Stieltjes constants, see A082633 and A086279. - Vaclav Kotesovec, Sep 10 2018
See Cully-Hugill & Trudgian, Theorem 2, for an explicit version of the asymptotic given above. - Charles R Greathouse IV, Nov 19 2019

Definition corrected by N. J. A. Sloane, May 25 2008

A319091 Decimal expansion of D, the coefficient of n in the asymptotic formula of Ramanujan for Sum_{k=1..n} (d(k)^2), where d(k) is the number of distinct divisors of k.

Original entry on oeis.org

4, 6, 0, 3, 2, 3, 3, 7, 2, 2, 5, 8, 7, 2, 1, 4, 3, 0, 3, 9, 3, 7, 6, 2, 0, 8, 6, 3, 8, 4, 4, 1, 8, 9, 7, 4, 7, 6, 3, 2, 1, 4, 9, 0, 3, 5, 3, 8, 7, 3, 9, 2, 2, 4, 0, 5, 8, 4, 2, 5, 0, 3, 4, 8, 4, 4, 5, 9, 0, 2, 6, 2, 9, 3, 2, 4, 0, 3, 2, 0, 7, 3, 8, 0, 1, 9, 8, 4, 8, 1, 0, 7, 6, 5, 9, 8, 5, 9, 9, 7, 3, 5, 6, 9, 5, 8

Offset: 0

Vaclav Kotesovec, Sep 10 2018

			0.4603233722587214303937620863844189747632149035387392240584250348445902629324...

Ramanujan's Papers, Some formulas in the analytic theory of numbers Messenger of Mathematics, XLV, 1916, 81-84, Formula (3).

Cf. A061502, A092742, A245074, A319090.

24*EulerGamma^3/Pi^2 - (432*Zeta'[2] /Pi^4+ 36/Pi^2)*EulerGamma^2 + (3456*Zeta'[2]^2/Pi^6 + 288*(Zeta'[2]-Zeta''[2])/Pi^4 + 24/Pi^2 - 72*StieltjesGamma[1]/Pi^2)*EulerGamma + StieltjesGamma[1]*(288*Zeta'[2]/Pi^4 + 24/Pi^2)-10368*Zeta'[2]^3/Pi^8 - 864*Zeta'[2]^2/Pi^6 + 1728*Zeta''[2] * Zeta'[2]/Pi^6 + 72*(Zeta''[2]-Zeta'[2])/Pi^4 - 48*Zeta'''[2]/Pi^4 + (12*StieltjesGamma[2] - 6)/Pi^2

D = 24*gamma^3/Pi^2 - (432*z1 /Pi^4+ 36/Pi^2)*gamma^2 + (3456*z1^2/Pi^6 + 288*(z1-z2)/Pi^4 + 24/Pi^2 - 72*g1/Pi^2)*gamma + g1*(288*z1/Pi^4 + 24/Pi^2)-10368*z1^3/Pi^8 - 864*z1^2/Pi^6 + 1728*z2*z1/Pi^6 + 72*(z2-z1)/Pi^4- 48*z3/Pi^4 + (12*g2-6)/Pi^2, where gamma is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and g1, g2 are the Stieltjes constants, see A082633 and A086279.

A320896 a(n) = Sum_{k=1..n} k * tau(k)^2, where tau is A000005.

Original entry on oeis.org

1, 9, 21, 57, 77, 173, 201, 329, 410, 570, 614, 1046, 1098, 1322, 1562, 1962, 2030, 2678, 2754, 3474, 3810, 4162, 4254, 5790, 6015, 6431, 6863, 7871, 7987, 9907, 10031, 11183, 11711, 12255, 12815, 15731, 15879, 16487, 17111, 19671, 19835, 22523, 22695, 24279

Offset: 1

Vaclav Kotesovec, Oct 23 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84.

Cf. A061502, A318755, A320897.

Cf. A000005, A006218, A143127, A319085, A320895.

Accumulate[Table[k*DivisorSigma[0, k]^2, {k, 1, 50}]]

a(n) = sum(k=1, n, k*numdiv(k)^2); \\ Michel Marcus, Oct 23 2018

a(n) ~ n^2 * (3*(Pi^6*(-1 - 24*g^2 + 32*g^3 + g*(8 - 96*s1) + 16*s1 + 16*s2) - 13824*z1^3 + 576*Pi^2*z1*((-1 + 8*g)*z1 + 4*z2) - 8*Pi^4*(3*(1 - 8*g + 24*g^2 - 16*s1)*z1 - 6*z2 + 48*g*z2 + 8*z3)) + 6*(Pi^6*(1 - 8*g + 24*g^2 - 16*s1) + 576*Pi^2*z1^2 - 24*Pi^4*(-z1 + 8*g*z1 + 2*z2))*log(n) + 6*((-1 + 8*g)*Pi^6 - 24*Pi^4*z1)*log(n)^2 + 4*Pi^6*log(n)^3) / (8*Pi^8), where g is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and s1, s2 are the Stieltjes constants, see A082633 and A086279.

A320897 a(n) = Sum_{k=1..n} k^2 * tau(k)^2, where tau is A000005.

Original entry on oeis.org

1, 17, 53, 197, 297, 873, 1069, 2093, 2822, 4422, 4906, 10090, 10766, 13902, 17502, 23902, 25058, 36722, 38166, 52566, 59622, 67366, 69482, 106346, 111971, 122787, 134451, 162675, 166039, 223639, 227483, 264347, 281771, 300267, 319867, 424843, 430319, 453423

Offset: 1

Vaclav Kotesovec, Oct 23 2018

nonn

In general, for m>=0, Sum_{k=1..n} k^m * tau(k)^2 ~ n^(m+1) * (log(n))^3 / ((m+1) * Pi^2).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84.

Cf. A061502, A318755, A320896.

Cf. A000005, A006218, A143127, A319085, A320895.

Accumulate[Table[k^2*DivisorSigma[0, k]^2, {k, 1, 50}]]

a(n) = sum(k=1, n, k^2*numdiv(k)^2); \\ Michel Marcus, Oct 23 2018

a(n) ~ n^3 * (2*Pi^6*(-1 + 12*g - 54*g^2 + 108*g^3 + 36*s1 - 324*g*s1 + 54*s2) - 93312*z1^3 + 2592*Pi^2*z1*(-z1 + 12*g*z1 + 6*z2) - 72*Pi^4*(z1 - 12*g*z1 + 54*g^2*z1 - 36*s1*z1 - 3*z2 + 36*g*z2 + 6*z3) + 6*(Pi^6*(1 - 12*g + 54*g^2 - 36*s1) + 1296*Pi^2*z1^2 - 36*Pi^4*(-z1 + 12*g*z1 + 3*z2))*log(n) + 9*((-1 + 12*g)*Pi^6 - 36*Pi^4*z1)*log(n)^2 + 9*Pi^6*log(n)^3) / (27*Pi^8), where g is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and s1, s2 are the Stieltjes constants, see A082633 and A086279.

A261508 Decimal expansion of -zeta'''(0).

Original entry on oeis.org

6, 0, 0, 4, 7, 1, 1, 1, 6, 6, 8, 6, 2, 2, 5, 4, 4, 4, 7, 7, 6, 1, 0, 6, 0, 8, 1, 3, 3, 6, 6, 3, 7, 5, 2, 8, 5, 4, 6, 1, 8, 0, 7, 6, 6, 8, 2, 9, 5, 9, 8, 0, 1, 3, 2, 8, 9, 3, 0, 8, 1, 5, 4, 1, 3, 0, 8, 6, 0, 4, 3, 3, 7, 2, 0, 7, 3, 4, 8, 5, 5, 4, 0, 2, 3, 9, 3, 8, 2, 5, 1, 9, 4, 6, 5, 6, 6, 7, 7, 5, 7, 3, 3, 5, 8

Offset: 1

Vaclav Kotesovec, Aug 22 2015

			6.004711166862254447761060813366375285461807668295980132893...

Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), pp. 223-232.
Eric Weisstein's World of Mathematics, Riemann Zeta Function
Eric Weisstein's World of Mathematics, Stieltjes Constants

Cf. A075700, A257549, A201995, A082633, A086279.

Maple
```
evalf(-Zeta(3, 0), 120);
```

RealDigits[3*Log[2*Pi]*StieltjesGamma[1] + 3*EulerGamma*StieltjesGamma[1] + 3/2*StieltjesGamma[2] - Zeta[3] - 1/2*Log[2*Pi]^3 - 1/8*Pi^2*Log[2*Pi] + 3/2*EulerGamma^2*Log[2*Pi] + EulerGamma^3, 10, 120][[1]]

-zeta'''(0) \\ Charles R Greathouse IV, Mar 10 2016

cp's OEIS Frontend

A073002 Decimal expansion of -zeta'(2) (the first derivative of the zeta function at 2).

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A201994 Decimal expansion of zeta''(2), the second derivative of the Riemann zeta function at 2.

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A061502 a(n) = Sum_{k<=n} tau(k)^2, where tau = number of divisors function A000005.

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A319091 Decimal expansion of D, the coefficient of n in the asymptotic formula of Ramanujan for Sum_{k=1..n} (d(k)^2), where d(k) is the number of distinct divisors of k.

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A320896 a(n) = Sum_{k=1..n} k * tau(k)^2, where tau is A000005.

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A320897 a(n) = Sum_{k=1..n} k^2 * tau(k)^2, where tau is A000005.

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A261508 Decimal expansion of -zeta'''(0).

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