scholarly journals Hardy’s function Z(t): Results and problems

2017 ◽  
Vol 296 (1) ◽  
pp. 104-114 ◽  
Author(s):  
Aleksandar Ivić
Keyword(s):  
Hardy’S Function

scholarly journals Average of Hardy’s function at Gram points

2021 ◽  
Author(s):  
Xiaodong Cao ◽  
Yoshio Tanigawa ◽  
Wenguang Zhai
Keyword(s):  
Hardy’S Function

scholarly journals On the integral of Hardy’s function

2004 ◽  
Vol 83 (1) ◽  
pp. 41-47 ◽  
Author(s):  
A. Ivić
Keyword(s):  
Hardy’S Function

On the integral of Hardy's function $ Z(t)$

2008 ◽  
Vol 72 (3) ◽  
pp. 429-478 ◽  
Author(s):  
M A Korolev
Keyword(s):  
Hardy’S Function

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

2010 ◽  
Vol Volume 33 ◽  
Author(s):  
Matti Jutila
Keyword(s):  
Entire Function ◽  
Zeta Function ◽  
Mellin Transform ◽  
Critical Line ◽  
Classical Estimate ◽  
International Audience ◽  
Hardy’S Function

International audience 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.

scholarly journals On the Mellin transforms of powers of Hardy's function.

2010 ◽  
Vol Volume 33 ◽  
Author(s):  
Aleksandar Ivić
Keyword(s):  
Mellin Transform ◽  
Mellin Transforms ◽  
International Audience ◽  
Power Moments ◽  
Hardy’S Function ◽  
Natural Boundaries

International audience Various properties of the Mellin transform function $$\mathcal{M}_k(s):= \int_1^{\infty} Z^k(x)x^{-s}\,dx$$ are investigated, where $$Z(t):=\zeta(\frac{1}{2}+it)\,\chi(\frac{1}{2}+it)^{-1/2},~~~~\zeta(s)=\chi(s)\zeta(1-s)$$ is Hardy's function. Connections with power moments of $|\zeta(\frac{1}{2}+it)|$ are established, and natural boundaries of $\mathcal{M}_k(s)$ are discussed.

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