L$-functions with the Lindel\"of and Riemann hypotheses and trivial zeros at even negative integers
DOI:
https://doi.org/10.51094/jxiv.620Keywords:
L-functions, Lindeloef hypothesis, Riemann's functional equation, Riemann hypothesis, trivial zerosAbstract
Let $q \ge 2$ be an integer, $\zeta (s)$ be the Riemann zeta function, and put $T_q (s) := (s+1)(1-s)^{-1} (q^{s+2}-1) \zeta (s+2) - 4 \pi^2 s^{-1} (1-s)^{-1}(q^{3-s}-1) \zeta (s-2)$. In the present paper, we show that the function $T_q (s)$ has Riemann's functional equation and its zeros only at the negative even integers and satisfies the Lindel\"of and Riemann hypotheses. In addition, we give functions satisfy Riemann's functional equation and an analogue of the Lindel\"of hypothesis but do not fulfill an analogue of the Riemann hypothesis.
Conflicts of Interest Disclosure
The authors have no conflicts of interest directly relevant to the content of this article.Downloads *Displays the aggregated results up to the previous day.
References
T. M. Apostol, Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, New York, 1976.
E. Bombieri and D. A. Hejhal, On the distribution of zeros of linear combinations of Euler products. Duke Math. J. 80 (1995), no. 3, 821–862.
H. Hamburger, ¨ Uber die Riemannsche Funktionalgleichung der ζ-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.
H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004.
A. A. Karatsuba and S. M. Voronin, The Riemann zeta-function. Translated from the Russian by Neal Koblitz. De Gruyter Expositions in Mathematics, 5. Walter de Gruyter & Co., Berlin, 1992.
J. Lagarias and M. Suzuki, The Riemann hypothesis for certain integrals of Eisenstein series. J. Number Theory 118 (2006), no. 1, 98–122.
A. Laurinˇcikas and R. Garunkˇstis, The Lerch zeta-function. Kluwer Academic Publishers, Dordrecht, 2002.
H. L. Montgomery and R. C. Vaughan, Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics, 97. Cambridge University Press, Cambridge, 2007.
T. Nakamura, On zeros of bilateral Hurwitz and periodic zeta and zeta star functions. Rocky Mountain Journal of Mathematics 53 (2023), no. 1, 157–176.
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, L-functions with Riemann’s functional equation and the Riemann hypothesis, The Quarterly Journal of Mathematics, 74 (2023), no 4, 1495-1504.
E. Saias and A. Weingartner, Zeros of Dirichlet series with periodic coefficients. Acta Arith. 140 (2009), no. 4, 335–344.
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.
Downloads
Posted
Submitted: 2024-02-13 06:05:39 UTC
Published: 2024-02-16 00:50:19 UTC — Updated on 2024-05-27 10:06:28 UTC
Versions
- 2024-05-27 10:06:28 UTC (2)
- 2024-02-16 00:50:19 UTC (1)
Reason(s) for revision
The title is changed. Section 1.2 and Some references are added.License
Copyright (c) 2024
Takashi Nakamura
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
