Mean value theorems for L-functions over prime polynomials for the rational function field

Author's manuscript. The published version is available via: DOI: 10.4064/aa161-4-4The first and second moments are established for the family of quadratic Dirichlet L-functions over the rational function field at the central point s=1/2, where the character χ is defined by the Legendre symbol for polynomials over finite fields and runs over all monic irreducible polynomials P of a given odd degree. Asymptotic formulae are derived for fixed finite fields when the degree of P is large. The first moment obtained here is the function field analogue of a result due to Jutila in the number field setting. The approach is based on classical analytical methods and relies on the use of the analogue of the approximate functional equation for these L-functions.National Science Foundation (NSF)Leverhulme TrustAir Force Office of Scientific Research (USAF

Average values of L-series for real characters in function fields

ArticleThis is the author accepted manuscript. The final version is available from Springer via the DOI in this record.We establish asymptotic formulae for the first and second moments of quadratic Dirichlet L–functions, at the centre of the critical strip, associated to the real quadratic function field k( √ P) and inert imaginary quadratic function field k( √ γP) with P being a monic irreducible polynomial over a fixed finite field Fq of odd cardinality q and γ a generator of F × q . We also study mean values for the class number and for the cardinality of the second K-group of maximal order of the associated fields for ramified imaginary, real, and inert imaginary quadratic function fields over Fq. One of the main novelties of this paper is that we compute the second moment of quadratic Dirichlet L-functions associated to monic irreducible polynomials. It is worth noting that the similar second moment over number fields is unknown. The second innovation of this paper comes from the fact that, if the cardinality of the ground field is even then the task of average L-functions in function fields is much harder and, in this paper, we are able to handle this strenuous case and establish several mean values results of L-functions over function fields.The first author was supported by an EPSRC-IHES William Hodge ´ Fellowship and by the EPSRC grant EP/K021132X/1. The second and third authors were supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP)(No. 2014001824)

Correction to ‘Shifted convolution and the Titchmarsh divisor problem over Fq[t]

PublishedCorrection to original article: Phil. Trans. R. Soc. A 373, 20140308 (28 April 2015; Published online 23 March 2015) (doi:10.1098/rsta.2014.0308). Two of the equations in the original article contained a typographical error. The author's accepted manuscript of the original article is available in this repository via: http://hdl.handle.net/10871/2062

Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices

This is the final version. Available on open access from IOP Publishing via the DOI in this recordData availability statement: No new data were created or analysed in this study.We investigate the joint moments of derivatives of characteristic polynomi als over the unitary symplectic group Sp(2N) and the orthogonal ensembles SO(2N) and O −(2N). We prove asymptotic formulae for the joint moments of the n1th and n2th derivatives of the characteristic polynomials for all three matrix ensembles. Our results give two explicit formulae for each of the leading order coefficients, one in terms of determinants of hypergeometric functions and the other as combinatorial sums over partitions. We use our results to put forward conjectures on the joint moments of derivatives of L-functions with symplectic and orthogonal symmetry.Leverhulme TrustEngineering and Physical Sciences Research Council (EPSRC

The first moment of L(12,χ) for real quadratic function fields

This is the author accepted manuscript. The final version is available from Instytut Matematyczny via the DOI in this recordIn this paper we use techniques first introduced by Florea to improve the asymptotic formula for the first moment of the quadratic Dirichlet L-functions over the rational function field, running over all monic, square-free polynomials of even degree at the central point. With some extra technical difficulties that do not appear in Florea’s paper, we prove that there is an extra main term of size gq 2g+2 3 , whilst bounding the error term by q g 2 (1+ǫ) .Leverhulme Trus

The importance of qualitative analytical chemistry in chemistry courses in Brazilian universities.

The results of a survey of institutions offering undergraduate studies, with the objective of evaluating the importance of Qualitative Analytical Chemistry for Chemistry courses in Brazil, are presented and discussed. Judging by the data, the content of the course of Qualitative Analytical Chemistry is considered by the Brazilian institutions offering undergraduate studies to be a body of knowledge essential for the formation of the chemist. This aspect is deemed valid for both baccalaureate and teaching license studies.29116817

The fourth moment of derivatives of Dirichlet L-functions in function fields

This is the final version. Available on open access from Springer via the DOI in this recordWe obtain the asymptotic main term of moments of arbitrary derivatives of L-functions in the function field setting. Specifically, we obtain the first, second, and mixed fourth moments. The average is taken over all non-trivial characters of a prime modulus Q ∈ Fq[T], and the asymptotic limit is as deg Q −→ ∞. This extends the work of Tamam who obtained the asymptotic main term of low moments of L-functions, without derivatives, in the function field setting. It is also the function field q-analogue of the work of Conrey, who obtained the fourth moment of derivatives of the Riemann zeta-functionLeverhulme TrustEngineering and Physical Sciences Research Council (EPSRC

Small zeros of Dirichlet L-functions of quadratic characters of prime modulus

This is the author accepted manuscript. The final version is available from Springer Verlag via the DOI in this recordIn this paper, we investigate the distribution of the imaginary parts of zeros near the real axis of Dirichlet L-functions associated to the quadratic characters χp(⋅)=(⋅|p) with p a prime number. Assuming the Generalized Riemann Hypothesis (GRH), we compute the one-level density for the zeros of this family of L-functions under the condition that the Fourier transform of the test function is supported on a closed subinterval of (−1,1). We also write down the ratios conjecture for this family of L-functions a la Conrey, Farmer and Zirnbauer and derive a conjecture for the one-level density which is consistent with the main theorem of this paper and with the Katz–Sarnak prediction and includes lower order terms. Following the methods of Özlük and Snyder, we prove that GRH implies L(12,χp)≠0 for at least 75% of the primes.London Mathematical SocietyLeverhulme Trus

Mean values of derivatives of L-functions in function fields: IV

This is the final version. Available on open access from the Korean Mathematical Society via the DOI in this recordIn this series, we investigate the calculation of mean values of derivatives of Dirichlet L-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields. The present paper generalizes the results obtained in the first paper. For µ ≥ 1 an integer, we compute the mean value of the µ-th derivative of quadratic Dirichlet L-functions over the rational function field. We obtain the full polynomial in the asymptotic formulae for these mean values where we can see the arithmetic dependence of the lower order terms that appears in the asymptotic expansion.Leverhulme TrustNational Research Foundation of Korea (NRF

The fourth power mean of Dirichlet L-functions in Fq [T]

This is the final version. Available on open access from Springer Verlag via the DOI in this recordWe prove results on moments of L-functions in the function field setting, where the moment averages are taken over primitive characters of modulus R, where R is a polynomial in Fq[T]. We consider the behaviour as deg R → ∞ and the cardinality of the finite field is fixed. Specifically, we obtain an exact formula for the second moment provided that R is square-full, an asymptotic formula for the second moment for any R, and an asymptotic formula for the fourth moment for any R. The fourth moment result is a function field analogue of Soundararajan’s result in the number field setting that improved upon a previous result by Heath-Brown. Both the second and fourth moment results extend work done by Tamam in the function field setting who focused on the case where R is prime. As a prerequisite for the fourth moment result, we obtain, for the special case of the divisor function, the function field analogue of Shiu’s generalised Brun-Titchmarsh theorem.Leverhulme TrustEngineering and Physical Sciences Research Council (EPSRC