Details

Sergio Venturini ^[a]

Non vanishing of Dirichlet series of completely multiplicative functions

Pages: 153-180
Received: 7 January 2019
Accepted in revised form: 30 September 2019
Mathematics Subject Classification (2010): Primary 11M06, 11M20; Secondary 30B40, 30B50.
Keywords: Riemann zeta function, Dirichlet series, Absolutely/completely monotone functions.
Authors address:
[a]: University of Bologna, Dipartimento di Matematica, Piazza di Porta S. Donato 5, 40127 Bologna, Italy

Abstract: Let \begin{equation*} L(s)=\sum_{n=1}^{+\infty}\dfrac{a(n)}{n^s} \end{equation*} be a Dirichlet series where \(a(n)\) is a bounded completely multiplicative function. We prove that if \(L(s)\) extends to a holomorphic function on the open half space \(\mathop{\rm Re} s >1-\delta\), \(\delta>0\) and \(L(1)=0\) then such a half space is a zero free region of the Riemann zeta function \(\zeta(s)\). Similar results are proven for completely multiplicative functions defined on the space of the ideals of the ring of the algebraic integers of a number field of finite degree.

References

[1]: T. M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR0434929
[2]: P. T. Bateman, A theorem of Ingham implying that Dirichlet's \(L\)-functions have no zeros with real part one, Enseign. Math. (2) 43 (1997), 281-284. MR1489887
[3]: S. Bernstein, Sur la définition et les propriétés des fonctions analytiques d'une variable réelle, Math. Ann. 75 (1914), 449-468. MR1511806
[4]: A. Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers généralisés, I, Acta Math. 68 (1937), 255-291. MR1577580
[5]: E. Hille, Analytic function theory. Vol. 1, Introduction to Higher Mathematics, Ginn and Company, Boston, 1959. MR0107692
[6]: A. E. Ingham, Note on Riemann's zeta-Function and Dirichlet's L-Functions, J. London Math. Soc. 5 (1930), 107-112. MR1574211
[7]: A. E. Ingham, The distribution of prime numbers, Cambridge Tracts in Mathematics and Mathematical Physics, No. 30, Stechert-Hafner, New York, 1964. MR0184920
[8]: J. Knopfmacher, Abstract analytic number theory, North-Holland Mathematical Library, 12, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., New York, 1975. MR0419383
[9]: S. Lang, Algebraic Number Theory, Grad. Texts in Math., 110, Springer-Verlag, 1994. MR1282723
[10]: H. L. Montgomery and R. C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Stud. Adv. Math., 97, Cambridge University Press, Cambridge, 2007. MR2378655
[11]: M. R. Murty and V. K. Murty, Non-vanishing of \(L\)-functions and applications, Progr. Math., 157, Birkhäuser Verlag, Basel, 1997. MR1482805
[12]: V. K. Murty, On the Sato-Tate conjecture, Number theory related to Fermat's last theorem (Cambridge, Mass., 1981), Progr. Math., 26, Birkhäuser, Boston, Mass., 1982, 195-205. MR0685296
[13]: R. Narasimhan, Une remarque sur \(\zeta (1+it)\), Enseign. Math. (2) 14 (1968), 189-191 (1969). MR0249373
[14]: W. Narkiewicz, Elementary and analytic theory of algebraic numbers, third ed., Springer Monogr. Math., Springer-Verlag, Berlin, 2004. MR2078267
[15]: J. Neukirch, Algebraic number theory, Translated from the 1992 German original and with a note by Norbert Schappacher, with a foreword by G. Harder, Grundlehren Math. Wiss., 322, Springer-Verlag, Berlin, 1999. MR1697859
[16]: D. J. Newman, A ''natural'' proof of the nonvanishing of \(L\)-series, A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math., 143, Amer. Math. Soc., Providence, RI, 1993, 495-498. MR1210536
[17]: A. P. Ogg, A remark on the Sato-Tate conjecture, Invent. Math. 9 (1969/1970), 198-200. MR0258835
[18]: M. Overholt, A course in analytic number theory, Grad. Stud. Math., 160, Amer. Math. Soc., Providence, RI, 2014. MR3290245
[19]: A. Pringsheim, Ueber Functionen, welche in gewissen Punkten endliche Differentialquotienten jeder endlichen Ordnung, aber keine Taylor'sche Reihenentwickelung besitzen, Math. Ann. 44 (1894), 41-56. MR1510831
[20]: G. Sansone and J. Gerretsen, Lectures on the theory of functions of a complex variable. I. Holomorphic functions, P. Noordhoff, Groningen, 1960. MR0113988
[21]: R. L. Schilling, R. Song and Z. Vondraček, Bernstein functions. Theory and applications, second ed., De Gruyter Stud. Math., 37, Walter de Gruyter & Co., Berlin, 2012. MR2978140
[22]: G. Shapiro, On the non-vanishing \(s=1\) of certain Dirichlet series, Amer. J. Math. 71 (1949), 621-626. MR0030552
[23]: J.-P. Serre, Zeta and \(L\) functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, 82-92. MR0194396
[24]: G. Tenenbaum, Introduction to analytic and probabilistic number theory, third ed., Translated from the 2008 French edition by Patrick D. F. Ion., Grad. Stud. Math., 163, Amer. Math. Soc., Providence, RI, 2015. MR3363366
[25]: D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, N. J., 1941. MR0005923
[26]: A. Wintner, The fundamental lemma in Dirichlet's theory of the arithmetical progressions, Amer. J. Math. 68 (1946), 285-292. MR0015422