L$-functions with the Lindel\"of and Riemann hypotheses and trivial zeros at even negative integers
DOI:
https://doi.org/10.51094/jxiv.620キーワード:
L-functions、 Lindeloef hypothesis、 Riemann's functional equation、 Riemann hypothesis、 trivial zeros抄録
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.
利益相反に関する開示
The authors have no conflicts of interest directly relevant to the content of this article.ダウンロード *前日までの集計結果を表示します
引用文献
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.
ダウンロード
公開済
投稿日時: 2024-02-13 06:05:39 UTC
公開日時: 2024-02-16 00:50:19 UTC — 2024-05-27 10:06:28 UTCに更新
バージョン
- 2024-05-27 10:06:28 UTC（2）
- 2024-02-16 00:50:19 UTC（1）
改版理由
The title is changed. Section 1.2 and Some references are added.ライセンス
Copyright（c）2024
Nakamura, Takashi
この作品は、Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Licenseの下でライセンスされています。
