Some remarks on the mean value of the Riemann zetafunction and other Dirichlet series. III [PDF]
Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1980 This is a sequel (Part II) to an earlier article with the same title. There are reasons to expect that the estimates proved in Part I without the factor $(\log\log H)^{-C}$ represent the real truth, and this is indeed proved in part II on the assumption that in the first estimate $2k$ is an integer.
K Ramachandra
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A UNIFIED METHOD FOR EVALUATING RIEMANN ZETA FUNCTIONS, DIRICHLET SERIES, ASSOCIATED CLAUSEN FUNCTIONS, OTHER ALLIED SERIES, AND NEW CLASSES OF INFINITE SERIES [PDF]
International Journal of Pure and Apllied Mathematics, 2014 Abstract: We have shown here for the first time that the completeness relation provides a simple unified theoretical framework for deriving different kinds of new recurrence formulae for Riemann Zeta Functions, Dirichlet series and Other Allied Series by selecting only different forms of complete set of orthonormal function (CSOF) in contrast to the ...
K.A. Acharya, K.J. Tej, A.K. Samanta
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Zeta-regularization of arithmetic sequences [PDF]
EPJ Web of Conferences, 2020 Is it possible to give a reasonable value to the infinite product 1 × 2 × 3 × · · · × n × · · · ? In other words, can we define some sort of convergence of the finite product 1 × 2 × 3 × · · · × n when n goes to infinity?
Allouche Jean-Paul
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Operator-valued zeta functions and Fourier analysis [PDF]
, 2019 The Riemann zeta function $\zeta(s)$ is defined as the infinite sum $\sum_{n=1}^\infty n^{-s}$, which converges when ${\rm Re}\,s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line ${\rm Re}\,s= \frac{1}{2}$. Thus,
Bender, Carl M., Brody, Dorje C
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On mean values of some zeta-functions in the critical strip [PDF]
, 2003 For a fixed integer $k\ge 3$ and fixed $1/2 1$ we consider $$ \int_1^T |\zeta(\sigma + it)|^{2k}dt = \sum_{n=1}^\infty d_k^2(n)n^{-2\sigma}T + R(k,\sigma;T), $$ where $R(k,\sigma;T) = o(T) (T\to\infty)$ is the error term in the above asymptotic formula.
Ivić, Aleksandar
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Heat-kernel coefficients of the Laplace operator on the D-dimensional ball [PDF]
, 1995 We present a very quick and powerful method for the calculation of heat-kernel coefficients. It makes use of rather common ideas, as integral representations of the spectral sum, Mellin transforms, non-trivial commutation of series and integrals and ...
Amsterdamski +36 more
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Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function [PDF]
, 2013 We survey some of the universality properties of the Riemann zeta function $\zeta(s)$ and then explain how to obtain a natural quantization of Voronin's universality theorem (and of its various extensions).
Herichi, Hafedh, Lapidus, Michel L.
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Universality and distribution of zeros and poles of some zeta functions [PDF]
, 2019 This paper studies zeta functions of the form $\sum_{n=1}^{\infty} \chi(n) n^{-s}$, with $\chi$ a completely multiplicative function taking only unimodular values. We denote by $\sigma(\chi)$ the infimum of those $\alpha$ such that the Dirichlet series $\
Seip, Kristian
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Representation growth and representation zeta functions of groups [PDF]
, 2012 We give a short introduction to the subject of representation growth and representation zeta functions of groups, omitting all proofs. Our focus is on results which are relevant to the study of arithmetic groups in semisimple algebraic groups, such as ...
Klopsch, Benjamin
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A Tapestry of Ideas with Ramanujan’s Formula Woven In
AxiomsZeta-functions play a fundamental role in many fields where there is a norm or a means to measure distance. They are usually given in the forms of Dirichlet series (additive), and they sometimes possess the Euler product (multiplicative) when the domain ...
Nianliang Wang +2 more
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