On some problems involving Hardy’s function
12
TL;DR: Some problems involving the classical Hardy function are discussed in this paper, where the odd moments of Z(t) and the distribution of its positive and negative values are discussed, as well as its distribution of odd moments.
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Abstract: Some problems involving the classical Hardy function $$
Z\left( t \right) = \zeta \left( {\frac{1}
{2} + it} \right)\left( {\chi \left( {\frac{1}
{2} + it} \right)} \right)^{ - {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} , \zeta \left( s \right) = \chi \left( s \right) \zeta \left( {1 - s} \right)
$$
, are discussed. In particular we discuss the odd moments of Z(t) and the distribution of its positive and negative values.
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Citations
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On the Riemann Zeta Function
TL;DR: In this paper, the authors give some number-theoretic applications of the theory of infinite series, based on the properties of the Riemann zeta function ς(s), which provides a link between number theory and real and complex analysis.
178
On the distribution of positive and negative values of Hardy's Z-function
Steven M. Gonek,Aleksandar Ivić +1 more
TL;DR: In this article, the authors investigated the distribution of positive and negative values of Hardy's function Z ( t ) : = ζ ( 1 2 + i t ) χ( 1 2+ i t + 1 2 ) − 1 / 2, ζ( s ) = χ ( s ) ζ 1 − s ).
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Non universality on the critical line
TL;DR: The authors showed that the Riemann zeta-function is not universal on the critical line by using the fact that the Hardy Z-function was real and some elementary considerations, which is a related result of Garunkstis and Steuding.
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On the distribution of positive and negative values of Hardy's $Z$-function
Steven M. Gonek,Aleksandar Ivić +1 more
TL;DR: In this article, the distribution of positive and negative values of Hardy's function was investigated, and it was shown that the distribution is similar to that of the Lebesgue measure.
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References
•Book
The Theory of the Riemann Zeta-Function
E. C. Titchmarsh,D. R. Heath-Brown +1 more
- 05 Feb 1987
TL;DR: The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it one of the most important tools in the study of prime numbers as mentioned in this paper.
4.2K
Random matrix theory and L-functions at s=1/2
Jon P Keating,Nina C Snaith +1 more
TL;DR: In this paper, the authors explore the link between the value distributions of the L-functions within these families at the central point s = 1/2 and those of the characteristic polynomials Z(U,θ) of matrices U with respect to averages over SO(2N) and USp(2Ns) at the corresponding point θ= 0, using techniques previously developed for U(N).
468
On the distribution of spacings between zeros of the zeta function
TL;DR: Etude numerique de la distribution des espacements des zeros de la fonction zeta de Riemann is presented in this article, where it is shown that the distribution of the zeros of the fonoord zeta can be described as follows:
459
•Posted Content
On the Riemann Zeta Function
TL;DR: In this paper, the authors give some number-theoretic applications of the theory of infinite series, based on the properties of the Riemann zeta function ς(s), which provides a link between number theory and real and complex analysis.
178