Some simplest integral equalities equivalent to the Riemann hypothesis

  • S. K. Sekatskii Laboratory of Biological Electron Microscopy, IPHYS, Ecole Polytechnique Fe ́de ́rale de Lausanne, Switzerland
  • S. Beltraminelli CERFIM, Research Center for Math. and Phys., Locarno, Switzerland)
Keywords: Riemann hypothesis, Integral equality

Abstract

UDC 511.3

We show that the following integral equalities are equivalent to the Riemann hypothesis for any real $a>0$ and any real $0<\epsilon<1,$ $\epsilon \neq 1$:\begin{gather*}\int\limits_{-\infty}^{\infty}\frac{\ln\left(\zeta\left(\dfrac{1}{2}+it\right)\right)}{a+it}\,dt=-2\pi\ln\frac{a+\dfrac{1}{2}}{a}, \\ \int\limits_{-\infty}^{\infty}\frac{\ln\left(\zeta\left(\dfrac{1}{2}+it\right)\right)}{(a+it)^\epsilon}\,dt=-\frac{2\pi}{1-\epsilon}\left(\left(a+\frac{1}{2}\right)^{1-\epsilon}-a^{1-\epsilon}\right).\end{gather*}

References

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Published
08.11.2022
How to Cite
SekatskiiS. K., and BeltraminelliS. “Some Simplest Integral Equalities Equivalent to the Riemann Hypothesis”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 9, Nov. 2022, pp. 1256 -3, doi:10.37863/umzh.v74i9.6222.
Section
Research articles