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# An estimate for the Mellin transform of powers of Hardy's function.

Abstract : We show that a certain modified Mellin transform $\mathcal M(s)$ of Hardy's function is an entire function. There are reasons to connect $\mathcal M(s)$ with the function $\zeta(2s-1/2)$, and then the orders of $\mathcal M(s)$ and $\zeta(s)$ should be comparable on the critical line. Indeed, an estimate for $\mathcal M(s)$ is proved which in the particular case of the critical line coincides with the classical estimate of the zeta-function.
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Cited literature [7 references]

https://hal.archives-ouvertes.fr/hal-01112386
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Submitted on : Monday, February 2, 2015 - 5:21:33 PM
Last modification on : Monday, March 28, 2022 - 8:14:08 AM
Long-term archiving on: : Wednesday, May 27, 2015 - 3:46:19 PM

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### Citation

Matti Jutila. An estimate for the Mellin transform of powers of Hardy's function.. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2010, Volume 33 - 2010, pp.23 - 31. ⟨10.46298/hrj.2010.169⟩. ⟨hal-01112386⟩

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