scholarly journals On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1254
Author(s):  
Xue Han ◽  
Xiaofei Yan ◽  
Deyu Zhang
Keyword(s):  
Asymptotic Formula ◽  
Fourier Coefficient ◽  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Mean Value ◽  
Symmetric Square ◽  
The Mean ◽  
Almost All

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

scholarly journals Large sums of Hecke eigenvalues of holomorphic cusp forms

2019 ◽  
Vol 31 (2) ◽  
pp. 403-417
Author(s):  
Youness Lamzouri
Keyword(s):  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Character Sums ◽  
Cusp Forms ◽  
Hecke Eigenvalues ◽  
Sign Change ◽  
Holomorphic Cusp Forms

AbstractLet f be a Hecke cusp form of weight k for the full modular group, and let {\{\lambda_{f}(n)\}_{n\geq 1}} be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of {\lambda_{f}(n)}, we investigate the range of x (in terms of k) for which there are cancellations in the sum {S_{f}(x)=\sum_{n\leq x}\lambda_{f}(n)}. We first show that {S_{f}(x)=o(x\log x)} implies that {\lambda_{f}(n)<0} for some {n\leq x}. We also prove that {S_{f}(x)=o(x\log x)} in the range {\log x/\log\log k\to\infty} assuming the Riemann hypothesis for {L(s,f)}, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which {S_{f}(x)\gg_{A}x\log x}, when {x=(\log k)^{A}}. Our results are {\mathrm{GL}_{2}} analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

Relationship between Mean Value and Fourier Coefficients of a Time-Varying Function with Half-Period Rests: Theory and Application

1994 ◽  
Vol 116 (1) ◽  
pp. 26-30
Author(s):  
Jun-Yao Yu
Keyword(s):  
Fourier Coefficient ◽  
Mathematical Theory ◽  
Fourier Coefficients ◽  
Mean Value ◽  
Time Function ◽  
Half Period ◽  
Time Varying ◽  
Ic Engine ◽  
The Mean ◽  
The Relationship

The Fourier coefficients are investigated for such a periodic time function whose magnitude keeps constant during the time of every half-period. In this case the relationship between the mean value and the Fourier coefficients is achieved using appropriate mathematical theory. It was applied successfully to the study of torsional harmonic excitations due to gas pressure in a four-cycle IC engine. A formula relating the truncated Fourier coefficient series to MIP has been established with very little error. The series simulation of Fourier coefficients is utilized in the determination and analysis of excitations.

ON THE MEAN VALUE OF THE INDEX OF COMPOSITION OF AN INTEGER II

2012 ◽  
Vol 09 (02) ◽  
pp. 431-445 ◽  
Author(s):  
DEYU ZHANG ◽  
MEIMEI LÜ ◽  
WENGUANG ZHAI
Keyword(s):  
Asymptotic Formula ◽  
Riemann Hypothesis ◽  
Error Term ◽  
Mean Value ◽  
Prime Factors ◽  
The Mean

For each integer n ≥ 2, let [Formula: see text] be the index of composition of n, where γ(n) ≔ ∏p∣np. The index of composition of an integer measures the multiplicity of its prime factors. In this paper, we obtain a new asymptotic formula of the sum ∑n≤xλ-k(n). Furthermore, we improve the error term under the Riemann Hypothesis.

scholarly journals A mean value of the representation function for the sum of two primes in arithmetic progressions

2017 ◽  
Vol 13 (04) ◽  
pp. 977-990 ◽  
Author(s):  
Yuta Suzuki
Keyword(s):  
Asymptotic Formula ◽  
Riemann Hypothesis ◽  
Mean Value ◽  
Arithmetic Progressions ◽  
Representation Function ◽  
Real Zeros ◽  
Generalized Riemann Hypothesis ◽  
Goldbach Problem ◽  
Primes In Arithmetic Progressions ◽  
The Mean

In this paper, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in arithmetic progressions. This is an improvement of the result of F. Rüppel in 2009, and a generalization of the result of A. Languasco and A. Zaccagnini concerning the ordinary Goldbach problem in 2012.

scholarly journals Two-Dimensional Divisor Problems Related to Symmetric L-Functions

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 359
Author(s):  
Jing Huang ◽  
Huafeng Liu ◽  
Fuxia Xu
Keyword(s):  
Asymptotic Formula ◽  
Cusp Form ◽  
Upper Bound ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Two Dimensional ◽  
Divisor Problems

In this paper, we study two-dimensional divisor problems of the Fourier coefficients of some automorphic product L-functions attached to the primitive holomorphic cusp form f(z) with weight k for the full modular group SL2(Z). Additionally, we establish the upper bound and the asymptotic formula for these divisor problems on average, respectively.

scholarly journals Mean representation number of integers as the sum of primes

2010 ◽  
Vol 200 ◽  
pp. 27-33 ◽  
Author(s):  
Gautami Bhowmik ◽  
Jan-Christoph Schlage-Puchta
Keyword(s):  
Riemann Hypothesis ◽  
Mean Value ◽  
Asymptotic Estimates ◽  
The Mean

AbstractAssuming the Riemann hypothesis, we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Ω-term, we show that our result is essentially the best possible.

A note on the mean value of L(1, χ) in the hyperelliptic ensemble

2014 ◽  
Vol 10 (04) ◽  
pp. 859-874 ◽  
Author(s):  
Hwanyup Jung
Keyword(s):  
Asymptotic Formula ◽  
Class Number ◽  
Mean Value ◽  
The Mean

In this paper, we establish an asymptotic formula for ∑D∈ℋ2g+2 L(1, χD) as g → ∞ (fixed odd q), where ℋ2g+2 is the subset of 𝔽q[T] consisting of monic square-free polynomials of degree 2g + 2. As an application, we obtain a formula for the average of the class number times the regulator of the associated rings [Formula: see text] when D is taken over ℋ2g+2 as g → ∞.

scholarly journals The hybrid mean value of Dedekind sums and two-term exponential sums

2016 ◽  
Vol 14 (1) ◽  
pp. 436-442
Author(s):  
Chang Leran ◽  
Li Xiaoxue
Keyword(s):  
Asymptotic Formula ◽  
Exponential Sums ◽  
Mean Value ◽  
Gauss Sums ◽  
Mean Value Theorem ◽  
Dedekind Sums ◽  
Hybrid Mean Value ◽  
The Mean ◽  
Interesting Identity

AbstractIn this paper, we use the mean value theorem of Dirichlet L-functions, the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the two-term exponential sums, and give an interesting identity and asymptotic formula for it.

scholarly journals Serum Adenosine Deaminase Values in Healthy People

2020 ◽  
Keyword(s):  
Adenosine Deaminase ◽  
Healthy Subjects ◽  
Reference Range ◽  
Mean Value ◽  
Normal Serum ◽  
Healthy People ◽  
Reference Ranges ◽  
Percentile Method ◽  
The Mean ◽  
Almost All

The purpose of this study is to determine the activity of serum adenosine deaminase (ADA) in healthy people, in connection with significant differences in published reference ranges from different authors. In our study, we examined 160 healthy subjects aged 18 to 84, of whom 64 were men and 96 women. We have determined serum adenosine deaminase levels using a method based on the ability of the enzyme adenosine deaminase to catalyze the deamination of adenosine to inosine and ammonia. The catalytic concentration is determined spectrophotometrically by the rate of reduction of NADH measured at 340 nm. We found that normal serum ADA values among our healthy subjects are higher than the recommended reference range for the method we use, namely below 18 U/l. Using the percentile method, we worked out the following reference ranges: for women 14.53 - 25.73 U/l and for men 18.46 – 27.50 U/l. For women, the mean value is 21.07 U/l, and for men 21.30 U/l. At 95% CI, the serum ADA values of almost all subjects included in the study are within the recommended and other authors range of 11.50 - 25.00 U/l.

scholarly journals Primes in Short Segments of Arithmetic Progressions

1998 ◽  
Vol 50 (3) ◽  
pp. 563-580 ◽  
Author(s):  
D. A. Goldston ◽  
C. Y. Yildirim
Keyword(s):  
Asymptotic Formula ◽  
Lower Bounds ◽  
Arithmetic Progression ◽  
Riemann Hypothesis ◽  
Total Variance ◽  
Correct Order ◽  
Asymptotic Results ◽  
Extended Range ◽  
Order Of Magnitude ◽  
Almost All

AbstractConsider the variance for the number of primes that are both in the interval [y,y + h] for y ∈ [x,2x] and in an arithmetic progression of modulus q. We study the total variance obtained by adding these variances over all the reduced residue classes modulo q. Assuming a strong form of the twin prime conjecture and the Riemann Hypothesis one can obtain an asymptotic formula for the total variance in the range when 1 ≤ h/q ≤ x1/2-∈ , for any ∈ > 0. We show that one can still obtain some weaker asymptotic results assuming the Generalized Riemann Hypothesis (GRH) in place of the twin prime conjecture. In their simplest form, our results are that on GRH the same asymptotic formula obtained with the twin prime conjecture is true for “almost all” q in the range 1 ≤ h/q ≤ x1/4-∈, that on averaging over q one obtains an asymptotic formula in the extended range 1 ≤ h/q ≤ x1/2-∈, and that there are lower bounds with the correct order of magnitude for all q in the range 1 ≤ h/q ≤ x1/3-∈.

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