ON THE SUM OF THE TWISTED EULER FUNCTION

2012 ◽  
Vol 08 (07) ◽  
pp. 1741-1761 ◽  
Author(s):  
JERZY KACZOROWSKI ◽  
KAZIMIERZ WIERTELAK
Keyword(s):  
Asymptotic Formula ◽  
Recent Work ◽  
Riemann Hypothesis ◽  
Error Term ◽  
Dirichlet Character ◽  
General Situation ◽  
Euler Function

We study an asymptotic formula for the sum of values of the Euler φ-function twisted by a real Dirichlet character. The error term is split into the arithmetic and the analytic part. The former is studied with minimal use of analytic tools in contrast to the latter, where the analysis depends heavily on the distribution of the non-trivial zeros of the corresponding Dirichlet L-function. The results of the present paper are an extension of a recent work by the authors, where the case of the classical Euler φ-function has been studied. The present, more general situation invites new technical difficulties. Not all of them can be successfully overcome. For instance, satisfactory omega results for the analytic part are proved in the case of an even Dirichlet character only. Nevertheless, a method providing good omega estimates for the arithmetic part as well as for the complete error term is developed. Moreover, it is noted that the Riemann Hypothesis for the involved Dirichlet L-function is equivalent to a sufficiently sharp estimation of the analytic part. This shows in particular that the arithmetic part can be much larger than the corresponding analytic part.

ON A NEW CIRCLE PROBLEM

2016 ◽  
Vol 103 (2) ◽  
pp. 231-249
Author(s):  
JUN FURUYA ◽  
MAKOTO MINAMIDE ◽  
YOSHIO TANIGAWA
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Dirichlet Character ◽  
Arithmetical Function ◽  
Circle Problem ◽  
The Riemann Zeta Function ◽  
Primitive Dirichlet Character ◽  
Riemann Zeta ◽  
The Mean ◽  
Voronoi Formula

We attempt to discuss a new circle problem. Let $\unicode[STIX]{x1D701}(s)$ denote the Riemann zeta-function $\sum _{n=1}^{\infty }n^{-s}$ ($\text{Re}\,s>1$) and $L(s,\unicode[STIX]{x1D712}_{4})$ the Dirichlet $L$-function $\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$ ($\text{Re}\,s>1$) with the primitive Dirichlet character mod 4. We shall define an arithmetical function $R_{(1,1)}(n)$ by the coefficient of the Dirichlet series $\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$$(\text{Re}\,s>1)$. This is an analogue of $r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$. In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for $\sum _{n\leq x}r(n)$. As a new problem, we deduce a ‘truncated Voronoï formula’ for the error term in the asymptotic formula for $\sum _{n\leq x}R_{(1,1)}(n)$. As a direct application, we show the mean square for the error term in our new problem.

scholarly journals Remark on Smith’s result on a divisor problem in arithmetic progressions

1985 ◽  
Vol 98 ◽  
pp. 37-42 ◽  
Author(s):  
Kohji Matsumoto
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Divisor Problem ◽  
Arithmetic Progressions ◽  
Euler Function

Let dk(n) be the number of the factorizations of n into k positive numbers. It is known that the following asymptotic formula holds: where r and q are co-prime integers with 0 < r < q, Pk is a polynomial of degree k − 1, φ(q) is the Euler function, and Δk(q; r) is the error term. (See Lavrik [3]).

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 The distribution of consecutive prime biases and sums of sawtooth random variables

2018 ◽  
Vol 168 (1) ◽  
pp. 149-169
Author(s):  
ROBERT J. LEMKE OLIVER ◽  
KANNAN SOUNDARARAJAN
Keyword(s):  
Fourier Transform ◽  
Asymptotic Formula ◽  
Recent Work ◽  
Error Term ◽  
Random Variables ◽  
Dominant Factor ◽  
Dedekind Sums ◽  
Secondary Factor ◽  
The Fourier Transform

AbstractIn recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to both the Fourier transform of classical Dedekind sums and the error term in the asymptotic formula for the sum of φ(n).

scholarly journals On the error term of a lattice counting problem, II

2020 ◽  
Vol 16 (05) ◽  
pp. 1153-1160
Author(s):  
Olivier Bordellès
Keyword(s):  
Asymptotic Formula ◽  
Riemann Hypothesis ◽  
Error Term ◽  
Exponential Sums ◽  
Main Term ◽  
Counting Problem ◽  
Lattice Problem

Under the Riemann Hypothesis, we improve the error term in the asymptotic formula related to the counting lattice problem studied in a first part of this work. The improvement comes from the use of Weyl’s bound for exponential sums of polynomials and a device due to Popov allowing us to get an improved main term in the sums of certain fractional parts of polynomials.

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.

XLIX.—Generalizations of a Problem of Pillai

1949 ◽  
Vol 62 (4) ◽  
pp. 460-469
Author(s):  
L. Mirsky
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Main Concern ◽  
Positive Integers ◽  
Infinite Set

I. Throughout this paper k1, …, k3 will denote s ≥ I fixed distinct positive integers. Some years ago Pillai (1936) found an asymptotic formula, with error term O(x/log x), for the number of positive integers n ≤ x such that n + k1, …, n + k3 are all square-free. I recently considered (Mirsky, 1947) the corresponding problem for r-free integers (i.e. integers not divisible by the rth power of any prime), and was able, in particular, to reduce the error term in Pillai's formula.Our present object is to discuss various generalizations and extensions of Pillai's problem. In all investigations below we shall be concerned with a set A of integers. This is any given, finite or infinite, set of integers greater than 1 and subject to certain additional restrictions which will be stated later. The elements of A will be called a-numbers, and the letter a will be reserved for them. A number which is not divisible by any a-number will be called A-free, and our main concern will be with the study of A-free numbers. Their additive properties have recently been investigated elsewhere (Mirsky, 1948), and some estimates obtained in that investigation will be quoted in the present paper.

scholarly journals The Mechanism of Branching of Three-Dimensional Periodic Orbits from the Plane

1983 ◽  
Vol 74 ◽  
pp. 213-224
Author(s):  
I.A. Robin ◽  
V.V. Markellos
Keyword(s):  
Recent Work ◽  
Periodic Orbits ◽  
Restricted Problem ◽  
Three Dimensional ◽  
General Situation ◽  
Orbital Symmetry ◽  
Resonant Orbits ◽  
Symmetry Properties ◽  
Elliptic Restricted Problem ◽  
Symmetric Periodic Orbits

AbstractA linearised treatment is presented of vertical bifurcations of symmetric periodic orbits(bifurcations of plane with three-dimensional orbits) in the circular restricted problem. Recent work on bifurcations from vertical-critical orbits (av = ±1) is extended to deal with the v more general situation of bifurcations from vertical self-resonant orbits (av = cos(2Πn/m) for integer m,n) and it is shown that in this more general case bifurcating families of three-dimensional orbits always occur in pairs, the orbital symmetry properties being governed by the evenness or oddness of the integer m. The applicability of the theory to the elliptic restricted problem is discussed.

scholarly journals The asymptotic behaviour of the mean-square of fractional part sums

2000 ◽  
Vol 43 (2) ◽  
pp. 309-323 ◽  
Author(s):  
Manfred Kühleitner ◽  
Werner Georg Nowak
Keyword(s):  
Asymptotic Formula ◽  
Asymptotic Behaviour ◽  
Error Term ◽  
Fractional Part ◽  
Mean Square ◽  
The Mean

AbstractIn this article we consider sums S(t) = Σnψ (tf(n/t)), where ψ denotes, essentially, the fractional part minus ½ f is a C4-function with f″ non-vanishing, and summation is extended over an interval of order t. For the mean-square of S(t), an asymptotic formula is established. If f is algebraic this can be sharpened by the indication of an error term.

THE EXCEPTIONAL SET IN THE POLYNOMIAL GOLDBACH PROBLEM

2011 ◽  
Vol 07 (03) ◽  
pp. 579-591 ◽  
Author(s):  
PAUL POLLACK
Keyword(s):  
Asymptotic Formula ◽  
Finite Field ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Natural Numbers ◽  
Exceptional Set ◽  
Goldbach Problem

For each natural number N, let R(N) denote the number of representations of N as a sum of two primes. Hardy and Littlewood proposed a plausible asymptotic formula for R(2N) and showed, under the assumption of the Riemann Hypothesis for Dirichlet L-functions, that the formula holds "on average" in a certain sense. From this they deduced (under ERH) that all but Oϵ(x1/2+ϵ) of the even natural numbers in [1, x] can be written as a sum of two primes. We generalize their results to the setting of polynomials over a finite field. Owing to Weil's Riemann Hypothesis, our results are unconditional.

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