# harmonic number

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##### 1: 25.16 Mathematical Applications
25.16.5 $H\left(s\right)=\sum_{n=1}^{\infty}\frac{H_{n}}{n^{s}},$
where $H_{n}$ is given by (25.11.33). …
25.16.13 $\sum_{n=1}^{\infty}\left(\frac{H_{n}}{n}\right)^{2}=\frac{17}{4}\zeta\left(4% \right),$
##### 2: 25.11 Hurwitz Zeta Function
25.11.32 $\int_{0}^{a}x^{n}\psi\left(x\right)\mathrm{d}x=(-1)^{n-1}\zeta'\left(-n\right)% +(-1)^{n}H_{n}\frac{B_{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{% }{n}{k}H_{k}\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}% {0.0pt}{}{n}{k}\zeta'\left(-k,a\right)a^{n-k},$ $n=1,2,\dots$, $\Re a>0$,
where $H_{n}$ are the harmonic numbers:
25.11.33 $H_{n}=\sum_{k=1}^{n}k^{-1}.$
##### 3: Errata
• Notation

Previously the notation $h(n)$ was used for the harmonic number $H_{n}$ (defined in (25.11.33)). The more widely used notation $H_{n}$ will now be used throughout the DLMF. In particular, this change was made in (25.11.32), (25.11.33), (25.16.5) and (25.16.13) (Suggested by Gergő Nemes on 2021-08-23).

• ##### 4: 1.2 Elementary Algebra
The geometric mean $G$ and harmonic mean $H$ of $n$ positive numbers $a_{1},a_{2},\dots,a_{n}$ are given by …
##### 5: Bibliography T
• J. W. Tanner and S. S. Wagstaff (1987) New congruences for the Bernoulli numbers. Math. Comp. 48 (177), pp. 341–350.
• N. M. Temme (1993) Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (3), pp. 233–243.
• A. Terras (1988) Harmonic Analysis on Symmetric Spaces and Applications. II. Springer-Verlag, Berlin.
• O. I. Tolstikhin and M. Matsuzawa (2001) Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.
• T. Ton-That, K. I. Gross, D. St. P. Richards, and P. J. Sally (Eds.) (1995) Representation Theory and Harmonic Analysis. Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.
• ##### 6: 1.9 Calculus of a Complex Variable
###### Mean Value Property
For $u(z)$ harmonic, …
##### 7: Bibliography H
• E. W. Hobson (1931) The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, London-New York.
• K. Horata (1989) An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.
• K. Horata (1991) On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.
• L. K. Hua (1963) Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.
• I. Huang and S. Huang (1999) Bernoulli numbers and polynomials via residues. J. Number Theory 76 (2), pp. 178–193.
• ##### 8: Bibliography M
• T. M. MacRobert (1967) Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.
• Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.
• G. Matviyenko (1993) On the evaluation of Bessel functions. Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.
• L. J. Mordell (1917) On the representation of numbers as a sum of $2r$ squares. Quarterly Journal of Math. 48, pp. 93–104.
• L. Moser and M. Wyman (1958a) Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33, pp. 133–146.
• ##### 9: Bibliography
• J. C. Adams and P. N. Swarztrauber (1997) SPHEREPACK 2.0: A Model Development Facility. NCAR Technical Note Technical Report TN-436-STR, National Center for Atmospheric Research.
• A. Adelberg (1996) Congruences of $p$-adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.
• H. Alzer (1997a) A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.
• H. Alzer (2000) Sharp bounds for the Bernoulli numbers. Arch. Math. (Basel) 74 (3), pp. 207–211.
• T. M. Apostol (2000) A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.
• ##### 10: 1.10 Functions of a Complex Variable
###### Harmonic Functions
If $u(z)$ is harmonic in $D$, $z_{0}\in D$, and $u(z)\leq u(z_{0})$ for all $z\in D$, then $u(z)$ is constant in $D$. Moreover, if $D$ is bounded and $u(z)$ is continuous on $\overline{D}$ and harmonic in $D$, then $u(z)$ is maximum at some point on $\partial D$. … Let $\alpha$ and $\beta$ be real or complex numbers that are not integers. … (The integer $k$ may be greater than one to allow for a finite number of zero factors.) …