Leonard Euler gave a general identity involving the harmonic number and the Riemann zeta function for natural numbers .

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A special case of this is the identity

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I first saw the above identity in Alex Youcis’s blog Abstract Nonsense and in course of further investigation, I was able to find several identities involving the Riemann zeta function and the harmonic numbers. While it is practically impossible to go through the entire mathematical literature to see if a formula is new or a rediscovery, in this post we shall see a few identities which are not given in the above links. Also if I encounter an identity elsewhere at any point of time, I have taken care to delete it from this post.

For the following relations hold.

**Entry 1**.

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**Entry 2**.

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**Entry 3**.

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**Entry 4**.

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**Entry 5**.

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**Entry 6**.

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**Entry 7**.

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**Entry a**. If then

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