Average Orders of Certain Arithmetical Functions
|
|
- Chad Griffith
- 5 years ago
- Views:
Transcription
1 Average Orders of Certain Arithmetical Functions Kaneenika Sinha July 26, 2006 Department of Mathematics and Statistics, Queen s University, Kingston, Ontario, Canada K7L 3N6, skaneen@mast.queensu.ca ABSTRACT. We consider the functions Mn), the maximum exponent of any prime power dividing n and mn), the minimum exponent of any prime power dividing n. The sums Mn) and mn) have been well investigated in the literature. In this note, we will improve known estimates of both the above sums under the assumption of the Riemann hypothesis. We will also obtain Ω-type estimates for these sums unconditionally. Mathematics Subject Classification 2000): Primary A5, Secondary A25. Introduction A natural number n 2 can be written uniquely as a product of prime numbers, We define and n = p α pα 2 2 pα k k, α i. Mn) = max{α, α 2,, α k } mn) = min{α, α 2,, α k }. We also define M) = m) =. The sums Mn) and mn) have been well investigated in the literature. In 969, Ivan Niven [0] proved that Mn) Bx, as x ) where B = + ). ζk) B can be evaluated to be.7052 approximately. He also showed that mn) = x + ζ ) 3 2 ζ3) x 2 + ox 2 ). 2) The aim of this note is to prove the following theorems: Research partially supported by a Queen s graduate fellowship.
2 Theorem Assuming the Riemann hypothesis, Mn) = Bx + Ox ɛ ). 3) Moreover, unconditionally, we have Mn) = Bx + Ωx 4 ). 4) Theorem 2 Assuming the Riemann hypothesis, we have where and γ i,k = 0 j k j i Also, unconditionally, we have mn) = x + p 7 i=2 ) k + j ζ + k + i Ai)x i + Ox ɛ ), 5) 2k m=k+ p m k+i 3k m=2k+2 A2) = γ 0,2, A3) = γ,2 + γ 0,3, A4) = γ,3 + γ 0,4, A5) = γ 2,3 + γ,4 + γ 0,5, A6) = γ 2,4 + γ,5 + γ 0,6, A7) = γ 3,4 + γ 2,5 + γ,6 + γ 0,7. mn) = x + 7 i=2 p m k+i Ai)x /i + Ωx 0 ). 6) Before proving these theorems, we would like to briefly review what is known about these sums. Perhaps unaware of Niven s work, in 975, C. W. Anderson proposed the following problem in the American Mathematical Monthly: Problem: Prove that lim x x where µm) is the Möbius function. Mn) = + m=2 µm) mm ), In response, T. Salat pointed out that the above sum had already been investigated by Niven in his 969 paper. A solution to Anderson s problem was also submitted by Ram Murty and Kumar Murty. In fact, they reported a mistake in the question and showed that the correct term on the right hand side should be µm) mm ), m=2 but this was not mentioned by the editor when responses to Anderson s problem were referenced in a later issue of the Monthly see [].) It is not difficult to see that this correct term is indeed equal to Niven s constant B. Since ) ζk) < for k 2, 2k 2 ),
3 the sum defining B in ) is absolutely convergent. Thus, by interchanging summations, we see that ) µm) µm) = ζk) m k = mm ). m=2 Niven s result was made effective by D. Suryanarayana and R. Sitaramachandrarao in 977. In [2], they proved that Mn) = Bx + x 2 exp c log 3/5 xlog log x) 5 ), 7) where c is an absolute positive constant. The main ingredient in their proof is a result of Walfisz proved in [4]), which we will state in the next section. They also prove that mn) = x + γ 0,2 x 2 + γ,2 + γ 0,3 )x 3 + γ,3 + γ 0,4 )x 4 m=2 + γ 2,3 + γ,4 + γ 0,5 )x 5 + Ox 6 ), 8) where γ i,k s are as defined in Theorem 2. Unaware of the results of [2], in 99, Hui Zhong Cao obtained an error estimate of Ox 2 log x) in [4]. This estimate is weaker than that obtained in [2]. Cao also provides an estimate for mn), but this is also not as sharp as equation 8). In the next two sections, we will estimate Mn). We will also refine some known estimates about mn) in the last section. The two main theorems proved in this note are Theorems and 2. 2 Preliminary results In this section, we will mention some results which are needed in this paper. For a fixed positive integer k 2, we define a k-free integer as a positive integer which is not divisible by the k-th power of any of its prime factors. Let S k x) denote the number of k-free integers less than or equal to x. It turns out that Mn) is directly related to the distribution of k-free numbers. By elementary number theory we know that for an integer k 2, S k x) = The direct connection between the error terms x ζk) + Ox k ). E k x) = S k x) x ζk) for k 2 and the error term in the sum Mn) is indicated in the following proposition, which is also mentioned in [0] and [2]: Proposition 3 where Mn) = Bx j = j E k x) + O), log x. log 2 3
4 Proof. We begin by observing that the maximum of {M), M2), Mn)} is log x log 2. Also, for k 2, the number of integers less than or equal to x such that Mn) = k is S k x) S k x). Thus, if we get that j = log x, log 2 Mn) = k )S k x) S k x)). j+ Since S j+ x) = x, the above sum is equal to jx + j k )S k x) which, on further simplification, is equal to Writing the above expression becomes This is equal to Since we get that Thus, k>j x + x jx S k x) = j Bx x k>j j ks k x), k= j S k x). x ζk) + Ox k ), ) ζk) ) ζk) ) < ζk) k>j j E k x). j E k x). ζk) < 2 k, Mn) = Bx 2 k = 2 j = O j E k x) + O). ). x While the trivial estimate for E k x) is Ox k ), substantially better estimates can be found, especially under the assumption of the Riemann hypothesis. We refer the reader to an excellent survey of Pappalardi [] on various improvements of this error term. In particular, the following results are relevant to this note: 4
5 In 963, Walfisz [4] proved that E k x) = x k exp ck 8/5 log 3/5 xlog log x) /5 ). 9) We advise the reader that Walfisz s theorem has not been recorded correctly in []. The exponent /5 in the third displayed formula on page 73 should be replaced by /5. The authors, in [2], prove 7) by applying Walfisz s result 9) for k = 2 and the trivial estimate Ox k ) for k 3 to the right hand side of Proposition 3. Clearly, any improvement in the estimates for E k x) will give a better error term for Mn). We now state two better estimates for E kx) which have been obtained by assuming the Riemann hypothesis. In 993, Jia [8] proved the following result: Proposition 4 Assuming the Riemann hypothesis, E 2 x) = Ox 7/54+ɛ ). The next result was proved in 98 by Montgomery and Vaughan [9]: Proposition 5 Assuming the Riemann hypothesis, for k 2, E k x) = Ox k+ +ɛ ) for any ɛ > 0. In 989, Graham and Pintz [5] obtained the following better estimate for E 3 x): Proposition 6 Assuming the Riemann hypothesis, 3 Proof of Theorem We recall, from Proposition 3, we get that By Proposition 4, E 3 x) = Ox 7/30 ). Mn) = Bx Moreover, by Proposition 5, for 3 k j, Thus, we get j E k x) + O). E 2 x) = Ox 7/54+ɛ ). E k x) = Ox 4 +ɛ ). Mn) = Bx + Ox 7/54+ɛ ). This proves equation 3), the first part of Theorem. In order to obtain equation 4), it suffices to show that E 2 x) + E 3 x) = Ωx 4 ). 0) This is because equation 9) gives us an unconditional estimate for E 4 x), which is sharper than Ox /4 ). Substituting the trivial estimate Ox /k ) for k 5, equation 9) for k = 4 and equation 0) in Proposition 3, we obtain equation 4). Equation 0) follows as a corollary to the following general result, which seems not to have been noticed before and is interesting in its own right: 5
6 Proposition 7 Let Sx) be a complex-valued function. Suppose that Sx) fs) = s dx xs+ has a pole at a complex number with real part θ. Then Sx) = Ωx θ ). Proof. The main idea of the proof of Proposition 7 is inspired by a technique followed by Vaidya in [3] to find out an Ω-type estimate for E k x) conditional upon either the Riemann hypothesis or the existence of a simple zero of ζks) on the line Res) = k/2 which is not a zero of ζs). Let us assume that Sx) = ox θ ). Then, for every ɛ > 0, we can find x 0 = x 0 ɛ) such that for x x 0, we have Sx) < ɛx θ. Let Sx) A for 0 < x < x 0. Thus, for s = σ + it with σ > θ, we have fs) < s A x θ dx + s ɛ = s A θ + s ɛ σ θ. x σ +θ dx Now, let us first consider the case when fs) has a simple pole at ρ = θ + it 0 with non-zero residue c. Then, as σ θ, t = t 0 ), we get that lim σ θ)fσ + it 0) = c. σ θ Thus, by the above inequality, we get that for every ɛ > 0, c = lim σ θ)fσ + it 0 ) σ θ lim σ θ = ɛ σ + it 0. { σ + it 0 σ θ) A θ + σ + it 0 ɛ This is not possible as c is non-zero. Thus, our assumption that Sx) = ox θ ) is false. Therefore, Sx) = Ωx θ ). The case when fs) has a pole of order 2 can also be dealt with by a similar analysis. This proves Proposition 7. In particular, if we let then it is not difficult to see that s Sx) = E 2 x) + E 3 x), Sx) ζs) dx = xs+ ζ2s) + ζs) ζ3s) ζs) ζ2) ζs) ζ3). ) Let ϱ = 4 + iγ be a pole of the right hand side of equation ). We prove the existence of such a pole by applying some zero-density arguments. A classical result of Selberg tells us that for T } 6
7 sufficiently large, the number of zeros of ζ2s) on the line segment joining 4 to 4 + it is at least AT log T for some A > 0. Such zeroes are the singularities of ζs) ζ2s) + ζs) ζ3s) ζs) ζ2) ζs) ζ3) unless possibly they are also zeros of ζs) or ζ3s). Since the real part of ϱ is /4, by the functional equation, we get a zero of ζs) where the real part of s is equal to 3/4. However, by a theorem of Carlson, see [6], Theorem 0.) we know that the number of zeros of ζs) up to height T with real part 3/4 is OT 3/4 log T ). Therefore, not every zero of ζ2s) is a zero of ζs) or ζ3s). Thus, the function s E 2 x) + E 3 x) x s+ dx will have at least one pole with real part /4. Thus, by applying Proposition 7 to the function Sx) = E 2 x) + E 3 x), we deduce equation 0). This concludes the proof of Theorem. 4 Proof of Theorem 2 In this section, we refine known estimates of the sum mn). We recall that equation 8) has been proved unconditionally in [2]. Once again, this estimate can be improved by assuming the Riemann hypothesis. Analogous to a k-free integer, a k-full integer is defined as a positive integer which is divisible by the k-th power of all its prime factors. We observe that analogous to Proposition 3), if L k x) denotes the number of k-full integers less than or equal to x, then mn) = x + L 2 x) + L 7 x) + Ox 8 ), 2) since j = Olog x) and k 8 L kx) = Ox /8 ) + Ox /9 log x). We now state some known results about the distribution of k-full integers. The following result has been proved in [7]: Proposition 8 For every k, L k x) = γ 0,k x /k + + γ k,k x /2k ) + k x), where the γ i,k s are as given after equation 8). Moreover, 3 x) = Ox ), 4 x) = Ox ), 5 x) = Ox 0 ) and 6 x) = Ox 2 ). The following conditional result proved in [5] in 200 gives an estimate for 2 x) : Proposition 9 Assuming the Riemann hypothesis, 2 x) = Ox ɛ ) for any ɛ > 0. We also recall the following unconditional Ω-type result of Balasubramanian, Ramachandra and Subbarao proved in [3]: Proposition 0 2 x) = Ωx /0 ). 7
8 We are now ready to prove Theorem 2: We had observed above in equation 2) that mn) = x + L 2 x) + + L 7 x) + Ox 8 ). Thus, by using Proposition 9 for k = 2, Proposition 8 for 3 k 6, and the elementary estimate for k = 7, we get that mn) = x + k x) = Ox k+ ) 7 i=2 Ai)x i + Ox ɛ ), where the Ai) s are as given above. We also observe that by applying equation 8), Proposition 8 and Proposition 0 to the right hand side of equation 2), we get that This proves Theorem 2. mn) = x + 5 Concluding remarks 7 Ai)x /i + Ωx 0 ). i=2 Thus, in this paper, we saw how closely the Riemann hypothesis is linked to the averages of exponents in factoring integers. The key ingredients in the proofs of Theorems and 2, as we saw in the last two sections, are the refined estimates of E k x) for k 2 obtained by assuming this ubiquitous conjecture of Riemann. We should note that these estimates were not available to Suryanarayana and Sitaramachandrarao when they wrote their paper [2] in 977. One would also like to know if the estimates for k x) for k 3 can be further improved under the assumption of the Riemann hypothesis or even some quasi-riemann hypothesis. Could one also obtain optimal error terms for the sums Mn) and mn)? We relegate these questions to future research. Acknowledgments. I would like to thank Professors E. Bertram and R. Murty for bringing this problem to my attention. I would also like to thank the referee for many helpful suggestions on a previous version of this paper. References [] C. W. Anderson, K. Murty, R. Murty and T. Salat, Solutions of advanced problems: Problem 605, The American Mathematical Monthly, Vol ), no. 0, pp [2] R. Balasubramanian and K. Ramachandra, On square-free numbers, Proceedings of the Ramanujan Centennial International Conference Annamalainagar, 987), RMS Publications, Ramanujan Math. Soc., Annamalainagar, 988, pp
9 [3] R. Balasubramanian, K. Ramachandra and M. V. Subbarao, On the error function in the asymptotic formula for the counting function of k-full numbers, Acta Arithmetica, Vol ), no. 2, pp [4] Hui Zhong Cao, The asymptotic formulas related to exponents in factoring integers, Math. Balkanica N.S.), Vol. 5 99), no. 2, pp [5] S. W. Graham and J. Pintz, The distribution of r-free numbers, Acta Math. Hungar., Vol ), no. -2, pp [6] H. Iwaniec and E. Kowalski, Analytic Number Theory, Vol ), AMS Colloquium Publications, American Math. Soc., Providence, Rhode Island. [7] A. Ivić and P. Shiu, The distribution of powerful integers, Illinois J. Math., Vol ), no. 4, pp [8] C. H. Jia, The distribution of square-free numbers, Sci. China Ser. A, 36, 993, No. 2, pp [9] H. L. Montgomery and R. C. Vaughan, The distribution of square-free numbers, Recent progress in analytic number theory, Vol., Durham 979), Academic Press, London-New York, 98, pp [0] I. Niven, Averages of exponents in factoring integers, Proceedings of the American Mathematical Society, Vol. 22, No. 2 August 969), pp [] F. Pappalardi, A survey on k-freeness, Ramanujan Mathematical Society Lecture Notes Series, No., edited by S. D. Adhikari, R. Balasubramaniam, K. Srinivas, 2004, pp [2] D. Suryanarayana and R. Sitaramachandrarao, On the maximum and minimum exponents in factoring integers, Arch. Math. Basel), Vol ), no. 3, pp [3] A. M. Vaidya, On the changes of sign of a certain error function connected with k-free integers, J. Indian Math. Soc, Vol. 32, 968, pp [4] A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, XV. VEB Deutscher Verlag der Wissenschaften, Berlin 963. [5] J. Wu, On the distribution of square-full integers, Arch. Math. Basel), Vol ), no. 3, pp
Some Arithmetic Functions Involving Exponential Divisors
2 3 47 6 23 Journal of Integer Sequences, Vol. 3 200, Article 0.3.7 Some Arithmetic Functions Involving Exponential Divisors Xiaodong Cao Department of Mathematics and Physics Beijing Institute of Petro-Chemical
More informationA remark on a conjecture of Chowla
J. Ramanujan Math. Soc. 33, No.2 (2018) 111 123 A remark on a conjecture of Chowla M. Ram Murty 1 and Akshaa Vatwani 2 1 Department of Mathematics, Queen s University, Kingston, Ontario K7L 3N6, Canada
More informationReciprocals of the Gcd-Sum Functions
2 3 47 6 23 Journal of Integer Sequences, Vol. 4 20, Article.8.3 Reciprocals of the Gcd-Sum Functions Shiqin Chen Experimental Center Linyi University Linyi, 276000, Shandong China shiqinchen200@63.com
More informationON THE DIVISOR FUNCTION IN SHORT INTERVALS
ON THE DIVISOR FUNCTION IN SHORT INTERVALS Danilo Bazzanella Dipartimento di Matematica, Politecnico di Torino, Italy danilo.bazzanella@polito.it Autor s version Published in Arch. Math. (Basel) 97 (2011),
More informationThe zeros of the derivative of the Riemann zeta function near the critical line
arxiv:math/07076v [mathnt] 5 Jan 007 The zeros of the derivative of the Riemann zeta function near the critical line Haseo Ki Department of Mathematics, Yonsei University, Seoul 0 749, Korea haseoyonseiackr
More informationSOME REMARKS ON ARTIN'S CONJECTURE
Canad. Math. Bull. Vol. 30 (1), 1987 SOME REMARKS ON ARTIN'S CONJECTURE BY M. RAM MURTY AND S. SR1NIVASAN ABSTRACT. It is a classical conjecture of E. Artin that any integer a > 1 which is not a perfect
More informationOn the number of special numbers
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 27, No. 3, June 207, pp. 423 430. DOI 0.007/s2044-06-0326-z On the number of special numbers KEVSER AKTAŞ and M RAM MURTY 2, Department of Mathematics Education,
More information1 i<j k (g ih j g j h i ) 0.
CONSECUTIVE PRIMES IN TUPLES WILLIAM D. BANKS, TRISTAN FREIBERG, AND CAROLINE L. TURNAGE-BUTTERBAUGH Abstract. In a stunning new advance towards the Prime k-tuple Conjecture, Maynard and Tao have shown
More informationUniform Distribution of Zeros of Dirichlet Series
Uniform Distribution of Zeros of Dirichlet Series Amir Akbary and M. Ram Murty Abstract We consider a class of Dirichlet series which is more general than the Selberg class. Dirichlet series in this class,
More informationUniform Distribution of Zeros of Dirichlet Series
Uniform Distribution of Zeros of Dirichlet Series Amir Akbary and M. Ram Murty Abstract We consider a class of Dirichlet series which is more general than the Selberg class. Dirichlet series in this class,
More informationOn the maximal exponent of the prime power divisor of integers
Acta Univ. Sapientiae, Mathematica, 7, 205 27 34 DOI: 0.55/ausm-205-0003 On the maimal eponent of the prime power divisor of integers Imre Kátai Faculty of Informatics Eötvös Loránd University Budapest,
More informationTHE LAPLACE TRANSFORM OF THE FOURTH MOMENT OF THE ZETA-FUNCTION. Aleksandar Ivić
THE LAPLACE TRANSFORM OF THE FOURTH MOMENT OF THE ZETA-FUNCTION Aleksandar Ivić Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. (2), 4 48. Abstract. The Laplace transform of ζ( 2 +ix) 4 is investigated,
More informationRiemann Zeta Function and Prime Number Distribution
Riemann Zeta Function and Prime Number Distribution Mingrui Xu June 2, 24 Contents Introduction 2 2 Definition of zeta function and Functional Equation 2 2. Definition and Euler Product....................
More informationBefore giving the detailed proof, we outline our strategy. Define the functions. for Re s > 1.
Chapter 7 The Prime number theorem for arithmetic progressions 7. The Prime number theorem We denote by π( the number of primes. We prove the Prime Number Theorem. Theorem 7.. We have π( as. log Before
More informationON PERMUTATION POLYNOMIALS OF PRESCRIBED SHAPE
ON PERMUTATION POLYNOMIALS OF PRESCRIBED SHAPE AMIR AKBARY, DRAGOS GHIOCA, AND QIANG WANG Abstract. We count permutation polynomials of F q which are sums of m + 2 monomials of prescribed degrees. This
More informationEvidence for the Riemann Hypothesis
Evidence for the Riemann Hypothesis Léo Agélas September 0, 014 Abstract Riemann Hypothesis (that all non-trivial zeros of the zeta function have real part one-half) is arguably the most important unsolved
More informationOn the representation of primes by polynomials (a survey of some recent results)
On the representation of primes by polynomials (a survey of some recent results) B.Z. Moroz 0. This survey article has appeared in: Proceedings of the Mathematical Institute of the Belarussian Academy
More information1. Introduction This paper investigates the properties of Ramanujan polynomials, which, for each k 0, the authors of [2] define to be
ZEROS OF RAMANUJAN POLYNOMIALS M. RAM MURTY, CHRIS SMYTH, AND ROB J. WANG Abstract. In this paper, we investigate the properties of Ramanujan polynomials, a family of reciprocal polynomials with real coefficients
More informationChapter 1. Introduction to prime number theory. 1.1 The Prime Number Theorem
Chapter 1 Introduction to prime number theory 1.1 The Prime Number Theorem In the first part of this course, we focus on the theory of prime numbers. We use the following notation: we write f( g( as if
More informationOn Some Mean Value Results for the Zeta-Function and a Divisor Problem
Filomat 3:8 (26), 235 2327 DOI.2298/FIL6835I Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Some Mean Value Results for the
More informationGAPS IN BINARY EXPANSIONS OF SOME ARITHMETIC FUNCTIONS, AND THE IRRATIONALITY OF THE EULER CONSTANT
Journal of Prime Research in Mathematics Vol. 8 202, 28-35 GAPS IN BINARY EXPANSIONS OF SOME ARITHMETIC FUNCTIONS, AND THE IRRATIONALITY OF THE EULER CONSTANT JORGE JIMÉNEZ URROZ, FLORIAN LUCA 2, MICHEL
More informationOn the Uniform Distribution of Certain Sequences
THE RAANUJAN JOURNAL, 7, 85 92, 2003 c 2003 Kluwer Academic Publishers. anufactured in The Netherlands. On the Uniform Distribution of Certain Sequences. RA URTY murty@mast.queensu.ca Department of athematics,
More informationINTEGERS DIVISIBLE BY THE SUM OF THEIR PRIME FACTORS
INTEGERS DIVISIBLE BY THE SUM OF THEIR PRIME FACTORS JEAN-MARIE DE KONINCK and FLORIAN LUCA Abstract. For each integer n 2, let β(n) be the sum of the distinct prime divisors of n and let B(x) stand for
More informationLINNIK S THEOREM MATH 613E UBC
LINNIK S THEOREM MATH 63E UBC FINAL REPORT BY COLIN WEIR AND TOPIC PRESENTED BY NICK HARLAND Abstract This report will describe in detail the proof of Linnik s theorem regarding the least prime in an arithmetic
More informationThe distribution of square-full integers
xl/2 The distribution of square-full integers D. SURYANARAYANA and 1%. SITA I~AI%IA CHANDRA_ I~AO Andhra University, Waltair, India 1. Introduction It is well-known that a positive integer n is called
More informationPRIME-REPRESENTING FUNCTIONS
Acta Math. Hungar., 128 (4) (2010), 307 314. DOI: 10.1007/s10474-010-9191-x First published online March 18, 2010 PRIME-REPRESENTING FUNCTIONS K. MATOMÄKI Department of Mathematics, University of Turu,
More informationFermat numbers and integers of the form a k + a l + p α
ACTA ARITHMETICA * (200*) Fermat numbers and integers of the form a k + a l + p α by Yong-Gao Chen (Nanjing), Rui Feng (Nanjing) and Nicolas Templier (Montpellier) 1. Introduction. In 1849, A. de Polignac
More informationSome Remarks on the Discrete Uncertainty Principle
Highly Composite: Papers in Number Theory, RMS-Lecture Notes Series No. 23, 2016, pp. 77 85. Some Remarks on the Discrete Uncertainty Principle M. Ram Murty Department of Mathematics, Queen s University,
More informationNORMAL GROWTH OF LARGE GROUPS
NORMAL GROWTH OF LARGE GROUPS Thomas W. Müller and Jan-Christoph Puchta Abstract. For a finitely generated group Γ, denote with s n(γ) the number of normal subgroups of index n. A. Lubotzky proved that
More informationNotes on the Riemann Zeta Function
Notes on the Riemann Zeta Function March 9, 5 The Zeta Function. Definition and Analyticity The Riemann zeta function is defined for Re(s) > as follows: ζ(s) = n n s. The fact that this function is analytic
More informationPARABOLAS INFILTRATING THE FORD CIRCLES SUZANNE C. HUTCHINSON THESIS
PARABOLAS INFILTRATING THE FORD CIRCLES BY SUZANNE C. HUTCHINSON THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics in the Graduate College of
More informationFirst, let me recall the formula I want to prove. Again, ψ is the function. ψ(x) = n<x
8.785: Analytic Number heory, MI, spring 007 (K.S. Kedlaya) von Mangoldt s formula In this unit, we derive von Mangoldt s formula estimating ψ(x) x in terms of the critical zeroes of the Riemann zeta function.
More informationYunhi Cho and Young-One Kim
Bull. Korean Math. Soc. 41 (2004), No. 1, pp. 27 43 ANALYTIC PROPERTIES OF THE LIMITS OF THE EVEN AND ODD HYPERPOWER SEQUENCES Yunhi Cho Young-One Kim Dedicated to the memory of the late professor Eulyong
More informationSOME MEAN VALUE THEOREMS FOR THE RIEMANN ZETA-FUNCTION AND DIRICHLET L-FUNCTIONS D.A. KAPTAN, Y. KARABULUT, C.Y. YILDIRIM 1.
SOME MEAN VAUE HEOREMS FOR HE RIEMANN ZEA-FUNCION AND DIRICHE -FUNCIONS D.A. KAPAN Y. KARABUU C.Y. YIDIRIM Dedicated to Professor Akio Fujii on the occasion of his retirement 1. INRODUCION he theory of
More informationTHE GENERALIZED ARTIN CONJECTURE AND ARITHMETIC ORBIFOLDS
THE GENERALIZED ARTIN CONJECTURE AND ARITHMETIC ORBIFOLDS M. RAM MURTY AND KATHLEEN L. PETERSEN Abstract. Let K be a number field with positive unit rank, and let O K denote the ring of integers of K.
More information(k), is put forward and explicated.
A Note on the Zero-Free Regions of the Zeta Function N. A. Carella, October, 2012 Abstract: This note contributes a new zero-free region of the zeta function ζ(s), s = σ + it, s C. The claimed zero-free
More informationOn the counting function for the generalized. Niven numbers
On the counting function for the generalized Niven numbers R. C. Daileda, Jessica Jou, Robert Lemke-Oliver, Elizabeth Rossolimo, Enrique Treviño Introduction Let q 2 be a fied integer and let f be an arbitrary
More informationOn a question of A. Schinzel: Omega estimates for a special type of arithmetic functions
Cent. Eur. J. Math. 11(3) 13 477-486 DOI: 1.478/s11533-1-143- Central European Journal of Mathematics On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions Research Article
More informationTHE HYPERBOLIC METRIC OF A RECTANGLE
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 401 407 THE HYPERBOLIC METRIC OF A RECTANGLE A. F. Beardon University of Cambridge, DPMMS, Centre for Mathematical Sciences Wilberforce
More informationO N P O S I T I V E N U M B E R S n F O R W H I C H Q(n) D I V I D E S F n
O N P O S I T I V E N U M B E R S n F O R W H I C H Q(n) D I V I D E S F n Florian Luca IMATE de la UNAM, Ap. Postal 61-3 (Xangari), CP 58 089, Morelia, Michoacan, Mexico e-mail: fluca@matmor.unain.inx
More informationTwin primes (seem to be) more random than primes
Twin primes (seem to be) more random than primes Richard P. Brent Australian National University and University of Newcastle 25 October 2014 Primes and twin primes Abstract Cramér s probabilistic model
More informationPRIME NUMBER THEOREM
PRIME NUMBER THEOREM RYAN LIU Abstract. Prime numbers have always been seen as the building blocks of all integers, but their behavior and distribution are often puzzling. The prime number theorem gives
More informationThe Zeros of the Derivative of the Riemann Zeta Function Near the Critical Line
Ki, H 008 The Zeros of the Derivative of the Riemann Zeta Function Near the Critical Line, International Mathematics Research Notices, Vol 008, Article ID rnn064, 3 pages doi:0093/imrn/rnn064 The Zeros
More informationThe Continuing Story of Zeta
The Continuing Story of Zeta Graham Everest, Christian Röttger and Tom Ward November 3, 2006. EULER S GHOST. We can only guess at the number of careers in mathematics which have been launched by the sheer
More informationRiemann s ζ-function
Int. J. Contemp. Math. Sciences, Vol. 4, 9, no. 9, 45-44 Riemann s ζ-function R. A. Mollin Department of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada, TN N4 URL: http://www.math.ucalgary.ca/
More informationTheorem 1.1 (Prime Number Theorem, Hadamard, de la Vallée Poussin, 1896). let π(x) denote the number of primes x. Then x as x. log x.
Chapter 1 Introduction 1.1 The Prime Number Theorem In this course, we focus on the theory of prime numbers. We use the following notation: we write f( g( as if lim f(/g( = 1, and denote by log the natural
More informationOn the discrepancy of circular sequences of reals
On the discrepancy of circular sequences of reals Fan Chung Ron Graham Abstract In this paper we study a refined measure of the discrepancy of sequences of real numbers in [0, ] on a circle C of circumference.
More informationOrder and chaos. Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA
Order and chaos Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA Perfect shuffles Suppose you take a deck of 52 cards, cut it in half, and perfectly shuffle it (with the bottom card staying
More informationSpecial values of derivatives of L-series and generalized Stieltjes constants
ACTA ARITHMETICA Online First version Special values of derivatives of L-series and generalized Stieltjes constants by M. Ram Murty and Siddhi Pathak (Kingston, ON To Professor Robert Tijdeman on the occasion
More informationMEAN VALUE THEOREMS AND THE ZEROS OF THE ZETA FUNCTION. S.M. Gonek University of Rochester
MEAN VALUE THEOREMS AND THE ZEROS OF THE ZETA FUNCTION S.M. Gonek University of Rochester June 1, 29/Graduate Workshop on Zeta functions, L-functions and their Applications 1 2 OUTLINE I. What is a mean
More informationNewton, Fermat, and Exactly Realizable Sequences
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.2 Newton, Fermat, and Exactly Realizable Sequences Bau-Sen Du Institute of Mathematics Academia Sinica Taipei 115 TAIWAN mabsdu@sinica.edu.tw
More informationOn the counting function for the generalized Niven numbers
On the counting function for the generalized Niven numbers Ryan Daileda, Jessica Jou, Robert Lemke-Oliver, Elizabeth Rossolimo, Enrique Treviño Abstract Given an integer base q 2 and a completely q-additive
More informationOn the Error Term for the Mean Value Associated With Dedekind Zeta-function of a Non-normal Cubic Field
Λ44ΩΛ6fi Ψ ο Vol.44, No.6 05ffμ ADVANCES IN MAHEMAICS(CHINA) Nov., 05 doi: 0.845/sxjz.04038b On the Error erm for the Mean Value Associated With Dedekind Zeta-function of a Non-normal Cubic Field SHI Sanying
More informationChapter 1. Introduction to prime number theory. 1.1 The Prime Number Theorem
Chapter 1 Introduction to prime number theory 1.1 The Prime Number Theorem In the first part of this course, we focus on the theory of prime numbers. We use the following notation: we write f g as if lim
More informationRIEMANN HYPOTHESIS: A NUMERICAL TREATMENT OF THE RIESZ AND HARDY-LITTLEWOOD WAVE
ALBANIAN JOURNAL OF MATHEMATICS Volume, Number, Pages 6 79 ISSN 90-5: (electronic version) RIEMANN HYPOTHESIS: A NUMERICAL TREATMENT OF THE RIESZ AND HARDY-LITTLEWOOD WAVE STEFANO BELTRAMINELLI AND DANILO
More informationMEAN SQUARE ESTIMATE FOR RELATIVELY SHORT EXPONENTIAL SUMS INVOLVING FOURIER COEFFICIENTS OF CUSP FORMS
Annales Academiæ Scientiarum Fennicæ athematica Volumen 40, 2015, 385 395 EAN SQUARE ESTIATE FOR RELATIVELY SHORT EXPONENTIAL SUS INVOLVING FOURIER COEFFICIENTS OF CUSP FORS Anne-aria Ernvall-Hytönen University
More informationarxiv:math/ v2 [math.nt] 10 Oct 2009
On certain arithmetic functions involving exonential divisors arxiv:math/0610274v2 [math.nt] 10 Oct 2009 László Tóth Pécs, Hungary Annales Univ. Sci. Budaest., Sect. Com., 24 2004, 285-294 Abstract The
More informationGOLDBACH S PROBLEMS ALEX RICE
GOLDBACH S PROBLEMS ALEX RICE Abstract. These are notes from the UGA Analysis and Arithmetic Combinatorics Learning Seminar from Fall 9, organized by John Doyle, eil Lyall, and Alex Rice. In these notes,
More informationDirichlet s Theorem. Calvin Lin Zhiwei. August 18, 2007
Dirichlet s Theorem Calvin Lin Zhiwei August 8, 2007 Abstract This paper provides a proof of Dirichlet s theorem, which states that when (m, a) =, there are infinitely many primes uch that p a (mod m).
More informationIntegers without large prime factors in short intervals and arithmetic progressions
ACTA ARITHMETICA XCI.3 (1999 Integers without large prime factors in short intervals and arithmetic progressions by Glyn Harman (Cardiff 1. Introduction. Let Ψ(x, u denote the number of integers up to
More informationOn Tornheim s double series
ACTA ARITHMETICA LXXV.2 (1996 On Tornheim s double series by James G. Huard (Buffalo, N.Y., Kenneth S. Williams (Ottawa, Ont. and Zhang Nan-Yue (Beijing 1. Introduction. We call the double infinite series
More informationA SURVEY ON k FREENESS
A SURVEY ON k FREENESS FRANCESCO PAPPALARDI Abstract. We say that an integer n is k free k 2 if for every prime p the valuation v p n < k. If f : N Z, we consider the enumerating function Sf k defined
More information1, for s = σ + it where σ, t R and σ > 1
DIRICHLET L-FUNCTIONS AND DEDEKIND ζ-functions FRIMPONG A. BAIDOO Abstract. We begin by introducing Dirichlet L-functions which we use to prove Dirichlet s theorem on arithmetic progressions. From there,
More informationarxiv: v22 [math.gm] 20 Sep 2018
O RIEMA HYPOTHESIS arxiv:96.464v [math.gm] Sep 8 RUIMIG ZHAG Abstract. In this work we present a proof to the celebrated Riemann hypothesis. In it we first apply the Plancherel theorem properties of the
More informationGeneralized Euler constants
Math. Proc. Camb. Phil. Soc. 2008, 45, Printed in the United Kingdom c 2008 Cambridge Philosophical Society Generalized Euler constants BY Harold G. Diamond AND Kevin Ford Department of Mathematics, University
More informationOn the Subsequence of Primes Having Prime Subscripts
3 47 6 3 Journal of Integer Sequences, Vol. (009, Article 09..3 On the Subsequence of Primes Having Prime Subscripts Kevin A. Broughan and A. Ross Barnett University of Waikato Hamilton, New Zealand kab@waikato.ac.nz
More informationPrime Number Theory and the Riemann Zeta-Function
5262589 - Recent Perspectives in Random Matrix Theory and Number Theory Prime Number Theory and the Riemann Zeta-Function D.R. Heath-Brown Primes An integer p N is said to be prime if p and there is no
More informationHarmonic sets and the harmonic prime number theorem
Harmonic sets and the harmonic prime number theorem Version: 9th September 2004 Kevin A. Broughan and Rory J. Casey University of Waikato, Hamilton, New Zealand E-mail: kab@waikato.ac.nz We restrict primes
More informationTHE DIRICHLET DIVISOR PROBLEM, BESSEL SERIES EXPANSIONS IN RAMANUJAN S LOST NOTEBOOK, AND RIESZ SUMS. Bruce Berndt
Title and Place THE DIRICHLET DIVISOR PROBLEM, BESSEL SERIES EXPANSIONS IN RAMANUJAN S LOST NOTEBOOK, AND RIESZ SUMS Bruce Berndt University of Illinois at Urbana-Champaign IN CELEBRATION OF THE 80TH BIRTHDAY
More informationNOTES ON ZHANG S PRIME GAPS PAPER
NOTES ON ZHANG S PRIME GAPS PAPER TERENCE TAO. Zhang s results For any natural number H, let P (H) denote the assertion that there are infinitely many pairs of distinct primes p, q with p q H; thus for
More informationLecture 1: Small Prime Gaps: From the Riemann Zeta-Function and Pair Correlation to the Circle Method. Daniel Goldston
Lecture 1: Small Prime Gaps: From the Riemann Zeta-Function and Pair Correlation to the Circle Method Daniel Goldston π(x): The number of primes x. The prime number theorem: π(x) x log x, as x. The average
More informationON THE CONSTANT IN BURGESS BOUND FOR THE NUMBER OF CONSECUTIVE RESIDUES OR NON-RESIDUES Kevin J. McGown
Functiones et Approximatio 462 (2012), 273 284 doi: 107169/facm/201246210 ON THE CONSTANT IN BURGESS BOUND FOR THE NUMBER OF CONSECUTIVE RESIDUES OR NON-RESIDUES Kevin J McGown Abstract: We give an explicit
More informationJournal of Number Theory
Journal of Number Theory 133 2013) 1809 1813 Contents lists available at SciVerse ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt The abc conjecture and non-wieferich primes in arithmetic
More informationCollatz cycles with few descents
ACTA ARITHMETICA XCII.2 (2000) Collatz cycles with few descents by T. Brox (Stuttgart) 1. Introduction. Let T : Z Z be the function defined by T (x) = x/2 if x is even, T (x) = (3x + 1)/2 if x is odd.
More informationON THE RESIDUE CLASSES OF π(n) MODULO t
ON THE RESIDUE CLASSES OF πn MODULO t Ping Ngai Chung Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts briancpn@mit.edu Shiyu Li 1 Department of Mathematics, University
More informationConstructive bounds for a Ramsey-type problem
Constructive bounds for a Ramsey-type problem Noga Alon Michael Krivelevich Abstract For every fixed integers r, s satisfying r < s there exists some ɛ = ɛ(r, s > 0 for which we construct explicitly an
More informationTHE DIOPHANTINE EQUATION P (x) = n! AND A RESULT OF M. OVERHOLT. Florian Luca Mathematical Institute, UNAM, Mexico
GLASNIK MATEMATIČKI Vol. 37(57(2002, 269 273 THE IOPHANTINE EQUATION P (x = n! AN A RESULT OF M. OVERHOLT Florian Luca Mathematical Institute, UNAM, Mexico Abstract. In this note, we show that the ABC-conjecture
More informationThe Degree of the Splitting Field of a Random Polynomial over a Finite Field
The Degree of the Splitting Field of a Random Polynomial over a Finite Field John D. Dixon and Daniel Panario School of Mathematics and Statistics Carleton University, Ottawa, Canada {jdixon,daniel}@math.carleton.ca
More information#A61 INTEGERS 14 (2014) SHORT EFFECTIVE INTERVALS CONTAINING PRIMES
#A6 INTEGERS 4 (24) SHORT EFFECTIVE INTERVALS CONTAINING PRIMES Habiba Kadiri Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Canada habiba.kadiri@uleth.ca Allysa
More informationEigenvalues of the Redheffer Matrix and Their Relation to the Mertens Function
Eigenvalues of the Redheffer Matrix and Their Relation to the Mertens Function Will Dana June 7, 205 Contents Introduction. Notations................................ 2 2 Number-Theoretic Background 2 2.
More informationThe primitive root theorem
The primitive root theorem Mar Steinberger First recall that if R is a ring, then a R is a unit if there exists b R with ab = ba = 1. The collection of all units in R is denoted R and forms a group under
More informationarxiv: v11 [math.gm] 30 Oct 2018
Noname manuscript No. will be inserted by the editor) Proof of the Riemann Hypothesis Jinzhu Han arxiv:76.99v [math.gm] 3 Oct 8 Received: date / Accepted: date Abstract In this article, we get two methods
More informationON SUMS OF PRIMES FROM BEATTY SEQUENCES. Angel V. Kumchev 1 Department of Mathematics, Towson University, Towson, MD , U.S.A.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A08 ON SUMS OF PRIMES FROM BEATTY SEQUENCES Angel V. Kumchev 1 Department of Mathematics, Towson University, Towson, MD 21252-0001,
More informationSome elementary explicit bounds for two mollifications of the Moebius function
Some elementary explicit bounds for two mollifications of the Moebius function O. Ramaré August 22, 213 Abstract We prove that the sum { /d 1+ε is bounded by, uniformly in x 1, r and ε >. We prove a similar
More informationFlat primes and thin primes
Flat primes and thin primes Kevin A. Broughan and Zhou Qizhi University of Waikato, Hamilton, New Zealand Version: 0th October 2008 E-mail: kab@waikato.ac.nz, qz49@waikato.ac.nz Flat primes and thin primes
More informationMAXIMAL PERIODS OF (EHRHART) QUASI-POLYNOMIALS
MAXIMAL PERIODS OF (EHRHART QUASI-POLYNOMIALS MATTHIAS BECK, STEVEN V. SAM, AND KEVIN M. WOODS Abstract. A quasi-polynomial is a function defined of the form q(k = c d (k k d + c d 1 (k k d 1 + + c 0(k,
More informationRank-one Twists of a Certain Elliptic Curve
Rank-one Twists of a Certain Elliptic Curve V. Vatsal University of Toronto 100 St. George Street Toronto M5S 1A1, Canada vatsal@math.toronto.edu June 18, 1999 Abstract The purpose of this note is to give
More informationEXTENDED RECIPROCAL ZETA FUNCTION AND AN ALTERNATE FORMULATION OF THE RIEMANN HYPOTHESIS. M. Aslam Chaudhry. Received May 18, 2007
Scientiae Mathematicae Japonicae Online, e-2009, 05 05 EXTENDED RECIPROCAL ZETA FUNCTION AND AN ALTERNATE FORMULATION OF THE RIEMANN HYPOTHESIS M. Aslam Chaudhry Received May 8, 2007 Abstract. We define
More informationOrder and chaos. Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA
Order and chaos Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA Perfect shuffles Suppose you take a deck of 52 cards, cut it in half, and perfectly shuffle it (with the bottom card staying
More informationA SHARP RESULT ON m-covers. Hao Pan and Zhi-Wei Sun
Proc. Amer. Math. Soc. 35(2007), no., 355 3520. A SHARP RESULT ON m-covers Hao Pan and Zhi-Wei Sun Abstract. Let A = a s + Z k s= be a finite system of arithmetic sequences which forms an m-cover of Z
More information1 x i. i=1 EVEN NUMBERS RAFAEL ARCE-NAZARIO, FRANCIS N. CASTRO, AND RAÚL FIGUEROA
Volume, Number 2, Pages 63 78 ISSN 75-0868 ON THE EQUATION n i= = IN DISTINCT ODD OR EVEN NUMBERS RAFAEL ARCE-NAZARIO, FRANCIS N. CASTRO, AND RAÚL FIGUEROA Abstract. In this paper we combine theoretical
More informationCOMPUTATION OF JACOBSTHAL S FUNCTION h(n) FOR n < 50.
MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages 000 000 S 0025-5718(XX)0000-0 COMPUTATION OF JACOBSTHAL S FUNCTION h(n) FOR n < 50. THOMAS R. HAGEDORN Abstract. Let j(n) denote the smallest positive
More informationHOW OFTEN IS EULER S TOTIENT A PERFECT POWER? 1. Introduction
HOW OFTEN IS EULER S TOTIENT A PERFECT POWER? PAUL POLLACK Abstract. Fix an integer k 2. We investigate the number of n x for which ϕn) is a perfect kth power. If we assume plausible conjectures on the
More informationLANDAU-SIEGEL ZEROS AND ZEROS OF THE DERIVATIVE OF THE RIEMANN ZETA FUNCTION DAVID W. FARMER AND HASEO KI
LANDAU-SIEGEL ZEROS AND ZEROS OF HE DERIVAIVE OF HE RIEMANN ZEA FUNCION DAVID W. FARMER AND HASEO KI Abstract. We show that if the derivative of the Riemann zeta function has sufficiently many zeros close
More informationPartitions into Values of a Polynomial
Partitions into Values of a Polynomial Ayla Gafni University of Rochester Connections for Women: Analytic Number Theory Mathematical Sciences Research Institute February 2, 2017 Partitions A partition
More informationON THE SPEISER EQUIVALENT FOR THE RIEMANN HYPOTHESIS. Extended Selberg class, derivatives of zeta-functions, zeros of zeta-functions
ON THE SPEISER EQUIVALENT FOR THE RIEMANN HYPOTHESIS RAIVYDAS ŠIMĖNAS Abstract. A. Speiser showed that the Riemann hypothesis is equivalent to the absence of non-trivial zeros of the derivative of the
More informationSieve theory and small gaps between primes: Introduction
Sieve theory and small gaps between primes: Introduction Andrew V. Sutherland MASSACHUSETTS INSTITUTE OF TECHNOLOGY (on behalf of D.H.J. Polymath) Explicit Methods in Number Theory MATHEMATISCHES FORSCHUNGSINSTITUT
More informationarxiv: v2 [math.nt] 30 May 2016
arxiv:408.049v [math.nt] 30 May 06 The distribution of -free numbers and the derivative of the Riemann zeta-function Xianchang Meng Abstract Under the Riemann Hypothesis, we connect the distribution of
More informationMinimal Paths and Cycles in Set Systems
Minimal Paths and Cycles in Set Systems Dhruv Mubayi Jacques Verstraëte July 9, 006 Abstract A minimal k-cycle is a family of sets A 0,..., A k 1 for which A i A j if and only if i = j or i and j are consecutive
More informationOn the Voronin s universality theorem for the Riemann zeta-function
On the Voronin s universality theorem for the Riemann zeta-function Ramūnas Garunštis Abstract. We present a slight modification of the Voronin s proof for the universality of the Riemann zeta-function.
More information