Mellin transform of Dirichlet \(L\)-functions with primitive character. (English) Zbl 1488.11124
Summary: In the paper, meromorphic continuation for the modified Mellin transform of Dirichlet \(L\)-functions with primitive character is obtained.
MSC:
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
References:
| [1] | A. Balčiuūnas. A transformation formula related to Dirichlet \(L\)-functions with principal character. Lietuvos Matematikos Rinkinys, 53(A): 13-18, 2012. · Zbl 1342.11073 |
| [2] | A. Balčiuūnas. Mellin transform of Dirichlet \(L\)-functions with principal character. Šiauliai Math. Semin., 8(16): 7-26, 2013. · Zbl 1302.11055 |
| [3] | A. Balčiuūnas and A. Laurinčikas. The Laplace transform of Dirichlet \(L\)-functions. Nonlinear Anal. Model. Control, 17(2): 127-138, 2012. · Zbl 1305.11071 |
| [4] | K. Chandrasekharan. Arithmetical Functions. Nauka, Moskva, 1945. |
| [5] | T. Estermann. On the representation of a number as the sum of two products. Proc. Lond. Math. Soc., 31: 123-133, 1930. http://dx.doi.org/10.1112/plms/s2-31.1.123. · JFM 56.0174.02 · doi:10.1112/plms/s2-31.1.123 |
| [6] | A. Ivič. The Mellin transform and the Riemann zeta-function. In W.G. Nowak and J. Shoißengeier (Eds.), Proc. Conf. on Elementary and Analytic Number Theory, pp. 112-127. Univ. Wienn and Univ. für Bodenkultur, Vienna, 1996. · Zbl 0880.11061 |
| [7] | A. Ivič. On some conjectures and results for the Riemann zeta-function and Heckeseries. Acta Arith., 99(2): 115-145, 2001. http://dx.doi.org/10.4064/aa99-2-2. · Zbl 0984.11043 · doi:10.4064/aa99-2-2 |
| [8] | A. Ivič. The Mellin transform of the square of Riemann’s zeta-function. Int. J. Number Theory, 1(1): 65-73, 2005. http://dx.doi.org/10.1142/S1793042105000042. · Zbl 1077.11060 · doi:10.1142/S1793042105000042 |
| [9] | A. Ivič. The Laplace and Mellin transforms of powers of the Riemann zeta-function. Int. J. Math. Anal., 1(2): 113-140, 2006. · Zbl 1138.11336 |
| [10] | A. Ivič, M. Jutila and Y. Motohashi. The Mellin transform of powers of the zeta-function. Acta Arith., 95(4): 305-342, 2000. · Zbl 0960.11039 |
| [11] | A. Ivič and Y. Motohashi. The mean square of the error term for the fourth moment of the zeta-function. Proc. Lond. Math. Soc. (3), 69(2): 309-329, 1994. · Zbl 0805.11060 · doi:10.1112/plms/s3-69.2.309 |
| [12] | M. Jutila. Lectures on a Method in the Theory of Exponential Sums. Tata Inst. Fund. Res. Lectures 80, Bombay, 1987. · Zbl 0671.10031 |
| [13] | M. Jutila. The Mellin transforms of the square of Riemann’s zeta-function. Period. Math. Hung., 42: 179-190, 2001. http://dx.doi.org/10.1023/A:1015213127383. · Zbl 1012.11082 · doi:10.1023/A:1015213127383 |
| [14] | M. Jutila. The Mellin transforms of the fourth power of Riemann’s zeta-function. In S.D. Adhikari, R. Balasubramanian and R. Srinivas (Eds.), Number Theory, volume 1 of Lect. Ser., pp. 15-29. Ramanujan Math. Soc., Mysore, 2005. · Zbl 1195.11109 |
| [15] | A. Laurinčikas. A growth estimate for the Mellin transform of the Riemann zeta-function. Math. Notes, 89(1-2): 82-92, 2011. http://dx.doi.org/10.1134/S0001434611010081. · Zbl 1290.11119 · doi:10.1134/S0001434611010081 |
| [16] | A. Laurinčikas. Mean square of the Mellin transform of the Riemann zeta- function. Integral Transforms Spec. Funct., 22(9): 667-679, 2011. http://dx.doi.org/10.1080/10652469.2010.536412. |
| [17] | A. Laurinčikas. The Mellin transform of the square of the Riemann zeta-function in the critical strip. Integral Transforms Spec. Funct., 22(7): 467-476, 2011. http://dx.doi.org/10.1080/10652469.2010.520458. · Zbl 1238.11079 |
| [18] | M. Lukkarinen. The Mellin Transform of the Square of Riemann’s Zeta-Function and Atkinson’s Formula, volume 140 of Ann. Acad. Scie. Fenn. Math. Diss. Suomalainen Tiedeakatemia, Helsinki, 2005. · Zbl 1073.11059 |
| [19] | Y. Motohashi. A relation between the Riemann zeta-function and the hyperbolic Laplacian. Ann. Sc. Norm. Super., Pisa, Cl. Sci., 22(4): 299-313, 1995. · Zbl 0831.11046 |
| [20] | Y. Motohashi. Spectral Theory of the Riemann Zeta-Function. Cambridge University Press, Cambridge, 1997. · Zbl 0878.11001 · doi:10.1017/CBO9780511983399 |
| [21] | E.C. Titchmarsh. The Theory of Functions. Oxford University Press, London, 1939. · JFM 65.0302.01 |
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