The size of the Lerch zeta-function at places symmetric with respect to the line ( s ) = 1 / 2

Ramūnas Garunkštis; Andrius Grigutis

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 1, page 25-37
  • ISSN: 0011-4642

Abstract

top
Let ζ ( s ) be the Riemann zeta-function. If t 6 . 8 and σ > 1 / 2 , then it is known that the inequality | ζ ( 1 - s ) | > | ζ ( s ) | is valid except at the zeros of ζ ( s ) . Here we investigate the Lerch zeta-function L ( λ , α , 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 λ = α it is still possible to obtain a certain version of the inequality | L ( λ , λ , 1 - s ¯ ) | > | L ( λ , λ , s ) | .

How to cite

top

Garunkštis, Ramūnas, and Grigutis, Andrius. "The size of the Lerch zeta-function at places symmetric with respect to the line $\Re (s)=1/2$." Czechoslovak Mathematical Journal 69.1 (2019): 25-37. <http://eudml.org/doc/294293>.

@article{Garunkštis2019,
abstract = {Let $\zeta (s)$ be the Riemann zeta-function. If $t\ge 6.8$ 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 $|L(\lambda ,\lambda ,1-\overline\{s\})|>|L(\lambda ,\lambda ,s)|$.},
author = {Garunkštis, Ramūnas, Grigutis, Andrius},
journal = {Czechoslovak Mathematical Journal},
keywords = {Lerch zeta-function; functional equation; zero distribution},
language = {eng},
number = {1},
pages = {25-37},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The size of the Lerch zeta-function at places symmetric with respect to the line $\Re (s)=1/2$},
url = {http://eudml.org/doc/294293},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Garunkštis, Ramūnas
AU - Grigutis, Andrius
TI - The size of the Lerch zeta-function at places symmetric with respect to the line $\Re (s)=1/2$
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 1
SP - 25
EP - 37
AB - Let $\zeta (s)$ be the Riemann zeta-function. If $t\ge 6.8$ 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 $|L(\lambda ,\lambda ,1-\overline{s})|>|L(\lambda ,\lambda ,s)|$.
LA - eng
KW - Lerch zeta-function; functional equation; zero distribution
UR - http://eudml.org/doc/294293
ER -

References

top
  1. Alzer, H., 10.1007/s00009-011-0128-6, Mediterr. J. Math. 9 (2012), 439-452. (2012) Zbl1329.11086MR2954501DOI10.1007/s00009-011-0128-6
  2. Apostol, T. M., 10.1007/978-1-4757-5579-4, Undergraduate Texts in Mathematics, Springer, New York (1976). (1976) Zbl0335.10001MR0434929DOI10.1007/978-1-4757-5579-4
  3. Berndt, B. C., 10.1215/ijm/1256053183, Ill. J. Math. 14 (1970), 244-258. (1970) Zbl0188.34701MR0268363DOI10.1215/ijm/1256053183
  4. Dixon, R. D., Schoenfeld, L., 10.1215/S0012-7094-66-03333-3, Duke Math. J. 33 (1966), 291-292. (1966) Zbl0154.04601MR0190103DOI10.1215/S0012-7094-66-03333-3
  5. Garunkštis, R., 10.1023/A:1016158125685, Lith. Math. J. 42 (2002), 140-145 and Liet. Mat. Rink. 42 2002 179-184. (2002) Zbl1026.11070MR1947632DOI10.1023/A:1016158125685
  6. Garunkštis, R., Grigutis, A., 10.1007/s00025-015-0486-7, Result. Math. 70 (2016), 271-281. (2016) Zbl06625537MR3535007DOI10.1007/s00025-015-0486-7
  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. (1999) Zbl0971.11048MR1738812
  8. Garunkštis, R., Laurinčikas, A., Steuding, J., 10.1007/s000130300005, Arch. Math. 80 (2003), 47-60. (2003) Zbl1040.11064MR1968287DOI10.1007/s000130300005
  9. Garunkštis, R., Steuding, J., 10.1524/anly.2002.22.1.1, Analysis, München 22 (2002), 1-12. (2002) Zbl1039.11055MR1899910DOI10.1524/anly.2002.22.1.1
  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. (2002) Zbl1044.11084MR1964850
  11. Garunkštis, R., 10.1007/s10986-017-9373-0, Lith. Math. J. 57 (2017), 433-440. (2017) Zbl06834723MR3736193DOI10.1007/s10986-017-9373-0
  12. Grigutis, A., 10.3846/13926292.2015.1119213, Math. Model. Anal. 20 (2015), 852-865. (2015) MR3427171DOI10.3846/13926292.2015.1119213
  13. Hinkkanen, A., 10.1080/17476939708815042, Complex Variables, Theory Appl. 34 (1997), 119-139. (1997) Zbl0905.30027MR1473594DOI10.1080/17476939708815042
  14. Lagarias, J., 10.4064/aa-89-3-217-234, Acta Arith. 89 (1999), 217-234 correction ibid. 116 2005 293-294. (1999) Zbl0928.11035MR1691852DOI10.4064/aa-89-3-217-234
  15. Laurinčikas, A., Garunkštis, R., 10.1007/978-94-017-6401-8, Kluwer Academic Publishers, Dordrecht (2002). (2002) Zbl1028.11052MR1979048DOI10.1007/978-94-017-6401-8
  16. Matiyasevich, Y., Saidak, F., Zvengrowski, P., 10.4064/aa166-2-4, Acta Arith. 166 (2014), 189-200. (2014) Zbl1319.11055MR3277049DOI10.4064/aa166-2-4
  17. Nazardonyavi, S., Yakubovich, S., 10.7153/jmi-07-16, J. Math. Inequal. 7 (2013), 167-174. (2013) Zbl1306.11067MR3099609DOI10.7153/jmi-07-16
  18. Saidak, F., Zvengrowski, P., On the modulus of the Riemann zeta function in the critical strip, Math. Slovaca 53 (2003), 145-172. (2003) Zbl1048.11069MR1986257
  19. Sondow, J., Dumitrescu, C., 10.1007/s10998-010-1037-3, Period. Math. Hung. 60 (2010), 37-40. (2010) Zbl1218.11079MR2629652DOI10.1007/s10998-010-1037-3
  20. Spira, R., 10.1215/S0012-7094-65-03223-0, Duke Math. J. 32 (1965), 247-250. (1965) Zbl0154.04501MR0176964DOI10.1215/S0012-7094-65-03223-0
  21. Spira, R., 10.2307/2005626, Math. Comput. 27 (1973), 379-385. (1973) Zbl0283.10022MR0326995DOI10.2307/2005626
  22. Spira, R., 10.1090/S0025-5718-1976-0409382-2, Math. Comput. 30 (1976), 863-866. (1976) Zbl0341.10034MR0409382DOI10.1090/S0025-5718-1976-0409382-2
  23. Titchmarsh, E. C., The Theory of Functions, Oxford University Press, Oxford (1939). (1939) Zbl65.0302.01MR3155290
  24. Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, Oxford Science Publications, Oxford University Press, New York (1986). (1986) Zbl0601.10026MR0882550
  25. Trudgian, T. S., 10.7153/jmi-09-65, J. Math. Inequal. 9 (2015), 795-798. (2015) Zbl06524374MR3345137DOI10.7153/jmi-09-65

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.