L-functions with Riemann's functional equation and the Riemann hypothesis
DOI:
https://doi.org/10.51094/jxiv.238Keywords:
L-functions, Riemann's functional equation, Riemann hypothesisAbstract
Let $\chi_4$ be the non-principal Dirichlet character mod $4$ and $L(s,\chi_4)$ be the Dirichlet $L$-function associated with $\chi_4$, and put $R(s):= s 4^{s} L(s+1,\chi_4) + \pi L(s-1,\chi_4)$. In the present paper, we show that the function $R(s)$ has the Riemann's functional equation and its zeros only at the negative even integers and complex numbers with real part $1/2$. We also give other $L$-functions that have the same property.
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References
T. M. Apostol, Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, New York, 1976.
A. Dixit and R. Gupta, Koshliakov zeta functions I: modular relations. Adv. Math. 393 (2021), Paper No. 108093, 41 pp. (arXiv:2108.00810).
H. Hamburger, "Uber die Riemannsche Funktionalgleichung der $zeta$-Funktion. (German) Math. Z. 10 (1921), no. 3-4, 240--254.
A. Ivi'{c}, The theory of Hardy's Z-function. Cambridge Tracts in Mathematics, 196. Cambridge University Press, Cambridge, 2013.
M. Knopp, On Dirichlet series satisfying Riemann's functional equation. Invent. Math. 117 (1994), no. 3, 361--372.
J. Lagarias and M. Suzuki, The Riemann hypothesis for certain integrals of Eisenstein series. J. Number Theory 118 (2006), no.~1, 98--122.
A. Laurinv{c}ikas and R. Garunkv{s}tis, The Lerch zeta-function. Kluwer Academic Publishers, Dordrecht, 2002.
T. Nakamura, On zeros of bilateral Hurwitz and periodic zeta and zeta star functions. to appear in Rocky Mountain Journal of Mathematics.
T. Nakamura, The functional equation and zeros on the critical line of the quadrilateral zeta function. J. Number Theory 233 (2022), 432--455 (arXiv:1910.09837).
T. Nakamura, Dirichlet series with periodic coefficients, Riemann's functional equation and real zeros of Dirichlet $L$-functions. to appear in Math. Slovaca.
T. Nakamura, On Lerch's formula and zeros of the quadrilateral zeta function, preprint, arXiv:2001.01981.
P. R. Taylor, On the Riemann zeta function, Quart. J. Oxford 19 (1945) 1--21.
E. C. Titchmarsh, The theory of the Riemann zeta-function, Second edition. Edited and with a preface by D. R. Heath-Brown. The Clarendon Press, Oxford University Press, New York, 1986.
Wikipedia, Riemann hypothesis url{https://en.wikipedia.org/wiki/Riemann_hypothesis}.
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Submitted: 2023-01-03 23:58:50 UTC
Published: 2023-01-05 06:52:57 UTC — Updated on 2023-01-16 02:55:32 UTC
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- 2023-02-01 04:40:52 UTC (3)
- 2023-01-16 02:55:32 UTC (2)
- 2023-01-05 06:52:57 UTC (1)
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Some tyops are corrected.License
Copyright (c) 2023
Takashi Nakamura
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