A073002 -id:A073002 - cp's OEIS frontend

A375503 Decimal expansion of the absolute value of zeta'(3/2), first derivative of the Riemann zeta function at s=3/2.

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

Offset: 1

R. J. Mathar, Aug 18 2024

			zeta'(3/2) = -3.93223973743110151070638857840601520269274355489257...

Cf. A073002 (at s=2), A244115 (s=3).

Maple
```
Zeta(1,3/2); evalf(%) ;
```

RealDigits[Zeta'[3/2], 10, 120][[1]] (* Amiram Eldar, Aug 19 2024 *)

A383224 Decimal expansion Sum_{p primes} log(p)^2*p^2/(p^2-1)^2.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Apr 27 2025

			0.8844818339635238851965361...

Bill Allombert, SumEulerLog procedure, Pari gp procedures.

Cf. A345364.

Zeta(2,2)/Zeta(2) -Zeta(1,2)^2/Zeta(2)^2 ; evalf(%) ; # R. J. Mathar, May 07 2025

RealDigits[(6 (-6 Zeta'[2]^2 + Pi^2 Zeta''[2]))/Pi^4, 10, 105][[1]]

/* Procedure by Bill Allombert */
default(realprecision, 105);
SumEulerLog(f,s=1,a=2,d=1)=
{
  my(p=variable(f));
  if(type(d)!="t_INT",error("incorrect type in SumEulerLog"));
  if (d<0,
    d=-d;
    for(i=1,d, f=deriv(f)*p);
    (-1)^d*intnum(t=1,[oo,log(2)*s],(t-1)^(d-1)*sumeulerrat(f,t*s,a))/gamma(d)
    ,d==0,
    sumeulerrat(f,s,a)
    ,d>0,
    my(S=0,v);
    my(prec=getlocalbitprec());
    f=subst(f,'p,1/p)+O(p^prec);
    for(i=1,d, f=intformal(f/p));
    v = valuation(f,p);
    f = truncate(f);
    for(i=v,prec/(v-1),
     S += polcoef(f,i)*derivnum(t=1,sumeulerrat(1/p,t*i*s,a),d));
    (-1)^d*S);
}
SumEulerLog(p^2/(p^2-1)^2,,,2)

Equals 6*(Pi^2*zeta''(2)-6*zeta'(2)^2)/Pi^4.
Equals 6*(Pi^2*zeta''(2)-6*(zeta[2]*(gamma + log(2*Pi) - 12*log(A)))^2)/Pi^4 where A is Glaisher-Kinkelin constant A074962.
Equals zeta''(2)/zeta(2)-zeta'(2)^2/zeta(2)^2 see A201994, A073002 and A013661.

A335007 Decimal expansion of 2*(gamma - zeta'(2)/zeta(2)) - 1, where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 19 2020

			1.2943533159921313340127529002042648668912832334937...

Eckford Cohen, The number of unitary divisors of an integer, The American Mathematical Monthly, Vol. 67, No. 9 (1960), pp. 879-880.

Cf. A001620 (gamma), A013661 (zeta(2)), A034444, A064608, A073002 (-zeta'(2)), A147533, A335006.

RealDigits[2*EulerGamma - 2*Zeta'[2]/Zeta[2] - 1, 10, 100][[1]]

2*Euler - 2*zeta'(2)/zeta(2) - 1 \\ Michel Marcus, May 19 2020

Equals lim_{k->oo} ((zeta(2)/k)*A064608(k) - log(k)) where A064608 is the partial sums of the number of unitary divisors (A034444).

Equals 2*A001620 + 2*A073002/A013661 - 1 = 2*A335006 - 1.

A343919 Decimal expansion of Sum_{k>=1} log(k+1/2)/k^2 .

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, May 04 2021

			1.435062...

Cf. A073002 (eps=0).

sumpos(k=1, log(k+1/2)/k^2) \\ Michel Marcus, May 04 2021

Sum_{k >=1 } log(k+eps)/k^2 = -Zeta'(2) - Sum_{i=1} (-eps)^i *Zeta(i+2)/i at eps=1/2, weighted sum over Riemann-Zeta.

A349522 Decimal expansion of Sum_{k>=2} 1/(k*log(k))^2.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Nov 20 2021

Theorem: Bertrand series Sum_{n>=2} 1/(n^q*log(n)^r) is convergent if q > 1.

Application for q = 2 with A201994 (r=-2), A073002 (r=-1), A013661 (r=0), A168218 (r=1), this sequence (r=2).

			0.6926058...

Wikipédia, Série de Bertrand (in French).
Wikipedia, Joseph Bertrand.

Cf. A013661, A073002, A168218, A201994.

sumpos(k=2, 1/(k*log(k))^2) \\ Michel Marcus, Nov 21 2021

Equals Sum_{k>=2} 1/(k*log(k))^2.

Equals Integral_{x>=2, y>=2} (zeta(x + y - 2) - 1) dx dy. - Amiram Eldar, Nov 21 2021

A349641 Decimal expansion of the Sum_{k>=2} 1/(k^3*log(k)).

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 24 2021

			Sum_{k>=2} 1/(k^3*log(k)) = 0.23799610019862130199...

Similar sequences: A013661, A002117, A073002, A244115, A168218.

(* following Jean-François Alcover's Mathematica program for A168218 *) digits = 110; NSum[ 1/(n^3*Log[n]), {n, 2, Infinity}, NSumTerms -> 500000, WorkingPrecision -> digits + 5, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 12}}] // RealDigits[#, 10, digits] & // First

intnum(x=3, [oo, log(3)], zeta(x)-1) \\ following Charles R Greathouse IV's program for A168218

sumpos(k=2, 1/(k^3*log(k))) \\ Michel Marcus, Nov 27 2021

Equals Integral_{s=3..oo} (zeta(s) - 1) ds.

A349770 a(n) = Sum_{d|n} usigma(d) * usigma(n/d).

Original entry on oeis.org

1, 6, 8, 19, 12, 48, 16, 48, 36, 72, 24, 152, 28, 96, 96, 113, 36, 216, 40, 228, 128, 144, 48, 384, 88, 168, 136, 304, 60, 576, 64, 258, 192, 216, 192, 684, 76, 240, 224, 576, 84, 768, 88, 456, 432, 288, 96, 904, 164, 528, 288, 532, 108, 816, 288, 768, 320, 360, 120, 1824

Offset: 1

Ilya Gutkovskiy, Nov 29 2021

Dirichlet convolution of A034448 with itself.

Cf. A034448, A034761, A322328, A328485, A343569.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; a[n_] := Sum[usigma[d] usigma[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]

Dirichlet g.f.: ( zeta(s) * zeta(s-1) / zeta(2*s-1) )^2.
Multiplicative with a(p^e) = e * (p^e + 1) + (p+1) * (p^e - 1)/(p-1). - Amiram Eldar, Nov 29 2021
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / zeta(3)^2 * (Pi^2 * log(n)/72 + gamma * Pi^2/36 - Pi^2/144 + zeta'(2)/6 - Pi^2 * zeta'(3)/(18*zeta(3))), where zeta(3) = A002117, zeta'(2) = -A073002, zeta'(3) = -A244115 and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 05 2021

A369885 Decimal expansion of Sum_{k>=1} log(k+1)/k^2.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 04 2024

			1.80075505600528299149660601421484318144566378381841...

Bernard Candelpergher and Marc-Antoine Coppo, A new class of identities involving Cauchy numbers, harmonic numbers and zeta values, The Ramanujan Journal, Vol. 27 (2012), pp. 305-328; alternative link.
István Mező, Problem 11793, Problems and Solutions, The American Mathematical Monthly, Vol. 121, No. 7 (2014), p. 648; A Series with Zetas, Solution to Problem 11793 by FAU Problem Solving Group, ibid., Vol. 123, No. 6 (2016), p. 620.
Michael Ian Shamos, A catalog of the real numbers (2011), p. 616.
Roberto Tauraso, Problem 11793.
Wikipedia, Harmonic number: Harmonic numbers for real and complex values.

Cf. A001008, A001620, A002805, A006232, A006233, A073002, A210593.

evalf(sum((-1)^(k+1)*Zeta(k)/(k-2), k = 3 .. infinity) - Zeta(1, 2), 120)

RealDigits[NIntegrate[HarmonicNumber[x]/x^2, {x, 1, Infinity}, WorkingPrecision -> 120]][[1]]

PARI
```
sumpos(k = 1, log(k+1)/k^2)
```

sumalt(k = 3, (-1)^(k+1) * zeta(k)/(k-2)) - zeta'(2)

Equals Integral_{x>=1} H(x)/x^2 dx, where H(x) is the harmonic number for real variable x (Shamos, 2011).
Equals -zeta'(2) + Sum_{k>=3} (-1)^(k+1)*zeta(k)/(k-2) (Mező, 2014).
Equals Sum_{k>=1} lambda(k)*H(k)/(k^2*k!) + 1 + zeta(3) - gamma * zeta(2), where lambda(k) = abs(A006232(k)/A006233(k)) is the n-th non-alternating Cauchy number, H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and gamma is Euler's constant (A001620) (Candelpergher and Coppo, 2012). - Amiram Eldar, Mar 18 2024

A386738 Decimal expansion of Integral_{x=0..1} {1/x}^4 dx, where {} denotes fractional part.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 01 2025

			0.14553289487913287197745596494722440166566646379514...

Ovidiu Furdui, Problem 3366, Crux Mathematicorum, Vol. 34, No. 6 (2008), pp. 362 and 365; Solution to Problem 3366, by Chip Curtis, ibid., Vol. 35, No. 6 (2009), pp. 403-405.
Huizeng Qin and Youmin Lu, Integrals of Fractional Parts and Some New Identities on Bernoulli Numbers, Int. J. Contemp. Math. Sciences, Vol. 6, No. 15 (2011), pp. 745-761.

Cf. A001620 (gamma), A002117, A061444, A073002, A306016.

Cf. A153810 (m=1), A345208 (m=2), A345208 (m=3), this constant (m=4).

RealDigits[Log[2*Pi] - 2*EulerGamma - 1/3 + (Zeta[3]/2 + Zeta'[2])/Zeta[2], 10, 120][[1]]

log(2*Pi) - 2*Euler - 1/3 + (zeta(3)/2 + zeta'(2))/zeta(2)

Equals log(2*Pi) - 2*gamma - 1/3 + 3*zeta(3)/Pi^2 + 6*zeta'(2)/Pi^2.

In general, for m >= 2, Integral_{x=0..1} {1/x}^m dx = log(2*Pi) - m*gamma/2 - 1/(m-1) - Sum_{k=1..floor((m-2)/2)} (-1)^k * (m!/(m-2*k-1)!) * zeta(2*k+1) / (2^(2*k+1) * Pi^(2*k)) + 2 * Sum_{k=1..floor((m-1)/2)} (-1)^(k-1) * (m!/(m-2*k)!) * zeta'(2*k) / (2*Pi)^(2*k).

cp's OEIS Frontend

A375503 Decimal expansion of the absolute value of zeta'(3/2), first derivative of the Riemann zeta function at s=3/2.

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A383224 Decimal expansion Sum_{p primes} log(p)^2*p^2/(p^2-1)^2.

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A335007 Decimal expansion of 2*(gamma - zeta'(2)/zeta(2)) - 1, where gamma is the Euler-Mascheroni constant.

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A343919 Decimal expansion of Sum_{k>=1} log(k+1/2)/k^2 .

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A349522 Decimal expansion of Sum_{k>=2} 1/(k*log(k))^2.

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A349641 Decimal expansion of the Sum_{k>=2} 1/(k^3*log(k)).

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A349770 a(n) = Sum_{d|n} usigma(d) * usigma(n/d).

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A369885 Decimal expansion of Sum_{k>=1} log(k+1)/k^2.

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