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Keywords:
Lerch zeta-function; functional equation; zero distribution
Summary:
Let $\zeta (s)$ be the Riemann zeta-function. If $t\geq 6.8$ and $\sigma >1/2$, then it is known that the inequality $|\zeta (1-s)|>|\zeta (s)|$ is valid except at the zeros of $\zeta (s)$. Here we investigate the Lerch zeta-function $L(\lambda ,\alpha ,s)$ which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the critical line. However, for equal parameters $\lambda =\alpha $ it is still possible to obtain a certain version of the inequality $|L(\lambda ,\lambda ,1-\overline {s})|>|L(\lambda ,\lambda ,s)|$.
References:
[1] Alzer, H.: Monotonicity properties of the Riemann zeta function. Mediterr. J. Math. 9 (2012), 439-452. DOI 10.1007/s00009-011-0128-6 | MR 2954501 | Zbl 1329.11086
[2] Apostol, T. M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, New York (1976). DOI 10.1007/978-1-4757-5579-4 | MR 0434929 | Zbl 0335.10001
[3] Berndt, B. C.: On the zeros of a class of Dirichlet series I. Ill. J. Math. 14 (1970), 244-258. DOI 10.1215/ijm/1256053183 | MR 0268363 | Zbl 0188.34701
[4] Dixon, R. D., Schoenfeld, L.: The size of the Riemann zeta-function at places symmetric with respect to the point ${1\over 2}$. Duke Math. J. 33 (1966), 291-292. DOI 10.1215/S0012-7094-66-03333-3 | MR 0190103 | Zbl 0154.04601
[5] Garunkštis, R.: On a positivity property of the Riemann $\xi$-function. Lith. Math. J. 42 (2002), 140-145 and Liet. Mat. Rink. 42 2002 179-184. DOI 10.1023/A:1016158125685 | MR 1947632 | Zbl 1026.11070
[6] Garunkštis, R., Grigutis, A.: The size of the Selberg zeta-function at places symmetric with respect to the line $ Re(s)=1/2$. Result. Math. 70 (2016), 271-281. DOI 10.1007/s00025-015-0486-7 | MR 3535007 | Zbl 06625537
[7] Garunkštis, R., Laurinčikas, A.: On zeros of the Lerch zeta-function. Number Theory and Its Applications. Proc. of the Conf. Held at the RIMS, Kyoto, 1997 Dev. Math. 2, Kluwer Academic Publishers, Dordrecht S. Kanemitsu et al. (1999), 129-143. MR 1738812 | Zbl 0971.11048
[8] Garunkštis, R., Laurinčikas, A., Steuding, J.: On the mean square of Lerch zeta-functions. Arch. Math. 80 (2003), 47-60. DOI 10.1007/s000130300005 | MR 1968287 | Zbl 1040.11064
[9] Garunkštis, R., Steuding, J.: On the zero distributions of Lerch zeta-functions. Analysis, München 22 (2002), 1-12. DOI 10.1524/anly.2002.22.1.1 | MR 1899910 | Zbl 1039.11055
[10] Garunkštis, R., Steuding, J.: Do Lerch zeta-functions satisfy the Lindelöf hypothesis?. Analytic and Probabilistic Methods in Number Theory. Proc. of the Third International Conf. in Honour of J. Kubilius, Palanga, 2001 TEV, Vilnius A. Dubickas, et al. (2002), 61-74. MR 1964850 | Zbl 1044.11084
[11] Garunkštis, R., unas, R. Tamoši\=: Symmetry of zeros of Lerch zeta-function for equal parameters. Lith. Math. J. 57 (2017), 433-440. DOI 10.1007/s10986-017-9373-0 | MR 3736193 | Zbl 06834723
[12] Grigutis, A., unas, D. Šiauči\=: On the modulus of the Selberg zeta-functions in the critical strip. Math. Model. Anal. 20 (2015), 852-865. DOI 10.3846/13926292.2015.1119213 | MR 3427171
[13] Hinkkanen, A.: On functions of bounded type. Complex Variables, Theory Appl. 34 (1997), 119-139. DOI 10.1080/17476939708815042 | MR 1473594 | Zbl 0905.30027
[14] Lagarias, J.: On a positivity property of the Riemann $\xi$-function. Acta Arith. 89 (1999), 217-234 correction ibid. 116 2005 293-294. DOI 10.4064/aa-89-3-217-234 | MR 1691852 | Zbl 0928.11035
[15] Laurinčikas, A., Garunkštis, R.: The Lerch Zeta-Function. Kluwer Academic Publishers, Dordrecht (2002). DOI 10.1007/978-94-017-6401-8 | MR 1979048 | Zbl 1028.11052
[16] Matiyasevich, Y., Saidak, F., Zvengrowski, P.: Horizontal monotonicity of the modulus of the zeta function, $L$-functions, and related functions. Acta Arith. 166 (2014), 189-200. DOI 10.4064/aa166-2-4 | MR 3277049 | Zbl 1319.11055
[17] Nazardonyavi, S., Yakubovich, S.: Another proof of Spira's inequality and its application to the Riemann hypothesis. J. Math. Inequal. 7 (2013), 167-174. DOI 10.7153/jmi-07-16 | MR 3099609 | Zbl 1306.11067
[18] Saidak, F., Zvengrowski, P.: On the modulus of the Riemann zeta function in the critical strip. Math. Slovaca 53 (2003), 145-172. MR 1986257 | Zbl 1048.11069
[19] Sondow, J., Dumitrescu, C.: A monotonicity property of Riemann's xi function and a reformulation of the Riemann hypothesis. Period. Math. Hung. 60 (2010), 37-40. DOI 10.1007/s10998-010-1037-3 | MR 2629652 | Zbl 1218.11079
[20] Spira, R.: An inequality for the Riemann zeta function. Duke Math. J. 32 (1965), 247-250. DOI 10.1215/S0012-7094-65-03223-0 | MR 0176964 | Zbl 0154.04501
[21] Spira, R.: Calculation of the Ramanujan $\tau $-Dirichlet series. Math. Comput. 27 (1973), 379-385. DOI 10.2307/2005626 | MR 0326995 | Zbl 0283.10022
[22] Spira, R.: Zeros of Hurwitz zeta functions. Math. Comput. 30 (1976), 863-866. DOI 10.1090/S0025-5718-1976-0409382-2 | MR 0409382 | Zbl 0341.10034
[23] Titchmarsh, E. C.: The Theory of Functions. Oxford University Press, Oxford (1939). MR 3155290 | Zbl 65.0302.01
[24] Titchmarsh, E. C.: The Theory of the Riemann Zeta-Function. Oxford Science Publications, Oxford University Press, New York (1986). MR 0882550 | Zbl 0601.10026
[25] Trudgian, T. S.: A short extension of two of Spira's results. J. Math. Inequal. 9 (2015), 795-798. DOI 10.7153/jmi-09-65 | MR 3345137 | Zbl 06524374
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