17 research outputs found
Special zeta Mahler functions
No full textIn 1969, I. Bernstein and S. Gelfand introduced an object, which is now
called the zeta Mahler function (ZMF, also zeta Mahler measure) and related to
the Mahler measure.
Here we discuss a family of ZMFs attached to the Laurent polynomials , where is real. We give explicit
formulae, present examples and establish properties for these ZMFs, such as an
RH-type phenomenon. Further, we explore relations with the Mahler measure.Comment: 26 page
On the modulo zeros of modular forms congruent to theta series
No full textFor a prime larger than , the Eisenstein series of weight has
some remarkable congruence properties modulo . Those imply, for example,
that the -invariants of its zeros (which are known to be real algebraic
numbers in the interval ), are at most quadratic over the field with
elements and are congruent modulo to the zeros of a certain truncated
hypergeometric series. In this paper we introduce "theta modular forms" of
weight for the full modular group as the modular forms for which the
first dim Fourier coefficients are identical to certain theta series. We
consider these theta modular forms for both the Jacobi theta series and the
theta series of the hexagonal lattice. We show that the -invariant of the
zeros of the theta modular forms for the Jacobi theta series are modulo all
in the ground field with elements. For the theta modular form of the
hexagonal lattice we show that its zeros are at most quadratic over the ground
field with elements. Furthermore, we show that these zeros in both cases
are congruent to the zeros of certain truncated hypergeometric functions.Comment: 18 page
On Lehmer's question for integer-valued polynomials
No full textWe solve a Lehmer-type question about the Mahler measure of integer-valued
polynomials.Comment: 9 page
On twisted period functions and Moments of a weighted mean square of Dirichlet L-functions on the critical line
No full textWe extend to Dirichlet L-functions associated with arbitrary primitive characters a range of objects and properties -- including Eisenstein series and period functions -- that were originally introduced and studied by Lewis and Zagier (2001), and later by Bettin and Conrey (2013) in the case of the Riemann zeta function, and more recently by Lewis and Zagier (2019) for odd real characters. These tools yield closed-form expressions for the moments of a measure defined via a weighted mean square of the L-function. These moments not only provide a complete characterization of the modulus of the L-function on the critical line, but also imply an infinite number of non-trivial positivity conditions valid for all primitive characters, real or not. The methods also involve a general form of an asymptotic formula based on the shifted Euler–Maclaurin summation formula, which may be of independent interest
On the zeros of odd weight Eisenstein series
No full textWe count the number of zeros of the holomorphic odd weight Eisenstein series
in all -translates of the standard fundamental
domain.Comment: 22 pages, 2 figures, 2 table
Critical points of modular forms
No full textWe count the number of critical points of a modular form with real Fourier coefficients in a -translate of the standard fundamental domain (with ). Whereas by the valence formula the (weighted) number of zeros of this modular form in is a constant only depending on its weight, we give a closed formula for this number of critical points in terms of those zeros of the modular form lying on the boundary of the value of and the weight. More generally, we indicate what can be said about the number of zeros of a quasimodular form.27 pages; final version with minor change
