Generalized Euler constants

Transcription

1 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 of Illinois, 409 West Green St., Urbana, IL 680, USA diamond@math.uiuc.edu, ford@math.uiuc.edu Received 9 March 2007; revised 7 July 2007 Abstract We study the distribution of a family {γp} of generalized Euler constants arising from integers sieved by finite sets of primes P. For P = P r, the set of the first r primes, γp r ep γ as r. Calculations suggest that γp r is monotonic in r, but we prove it is not. Also, we show a connection between the distribution of γp r ep γ and the Riemann hypothesis.. Introduction Euler s constant γ = also known as the Euler-Mascheroni constant reflects a subtle multiplicative connection between Lebesgue measure and the counting measure of the positive integers and appears in many contets in mathematics see e.g. the recent monograph [4]. Here we study a class of analogues involving sieved sets of integers and investigate some possible monotonicities. As a first eample, consider the sum of reciprocals of odd integers up to a point : we have and we take n n odd n = n 2 n n /2 { γ := lim n n odd More generally, if P represents a finite set of primes, let {, if n, p P p =, P n:= 0, else, n = γ + log 2 log + + o, 2 2 n } 2 log = γ + log 2. 2 and δ P := lim P n. A simple argument shows that δ P = p P /p and that the generalized Euler constant { P n } γp:= lim δ P log n n eists. We shall investigate the distribution of values of γp for various prime sets P. We begin by indicating two further representations of γp. First, a small Abelian argument shows that it is the Research of the second author supported by National Science Foundation grant DMS n

2 2 HAROLD G. DIAMOND AND KEVIN FORD constant term in the Laurent series about of the Dirichlet series P nn s = ζs p s, p P where ζ denotes the Riemann zeta function. That is, γp = lim s { ζs p P For a second representation, take P = p P p. We have n P n n = n n p s δ P }. s d n,p µd = d P µd d m /d m = d P µd d log/d + γ + Od/ = δ P log d P µd log d d + γδ P + o as, where µ is the Möbius function. If we apply the Dirichlet convolution identity µ log = Λ µ, where Λ is the von Mangoldt function, we find that Thus we have d P µd log d d = ab P Λaµb ab 2 γp = p P = p P PROPOSITION. Let P be any finite set of primes. Then p p { γ + p P b P/p µb b }. p = δ P We remark that this formula also can be deduced from by an easy manipulation. It is natural to inquire about the spectrum of values G = {γp: P is a finite set of primes}. In particular, what is Γ:= inf G? The closure of G is simple to describe in terms of Γ. PROPOSITION 2. The set G is dense in [Γ,. Proof. Suppose > Γ and let P be a finite set of primes with γp <. Put c = γp p. p P p P p. Let y be large and let P y be the union of P and the primes in y, e c y]. By 2, the well-known Mertens estimates, and the prime number theorem, γp y = log y + O γ + p c + log y log y p + c + O/ log y. p P p P Therefore, lim y γp y = and the proof is complete.

3 Generalized Euler constants 3 In case P consists of the first r primes {p,..., p r }, we replace P by r in the preceding notation, and let γ r represent the generalized Euler constant for the integers sieved by the first r primes. Also define γ 0 = γ = γ. These special values play an important role in our theory of generalized Euler constants. Table. Some Gamma Values truncated γ = γ = γ = γ 2 = γ 3 = γ 2 = γ 2 = γ 22 = γ 32 = γ 3 = γ 3 = γ 23 = γ 33 = γ 4 = γ 4 = γ 24 = γ 34 = γ 5 = γ 5 = γ 25 = γ 35 = γ 6 = γ 6 = γ 26 = γ 36 = γ 7 = γ 7 = γ 27 = γ 37 = γ 8 = γ 8 = γ 28 = γ 38 = γ 9 = γ 9 = γ 29 = γ 39 = γ 0 = γ 20 = γ 30 = γ 40 = e γ = The net result will be proved in Section 2. THEOREM. Let P be a finite set of primes. For some r, 0 r #P, we have γp γ r. Consequently, Γ = inf r 0 γ r. Applying Mertens well-known formulas for sums and products of primes to 2, we find that 3 γ r e γ r { r + O} e γ r. In particular, Γ e γ and G is dense in [e γ,. Values of γ r for all r with p r 0 9 were computed to high precision using PARI/GP. For all such r, γ r > e γ and γ r+ < γ r. It is natural to ask if these trends persist. That is, Is the sequence of γ r s indeed decreasing for all r? 2 If the γ r s oscillate, are any of them smaller than e γ, i.e., is Γ < e γ? We shall show that the answer to is No and the answer to 2 is No or Yes depending on whether the Riemann Hypothesis RH is true or false. THEOREM 2. There are infinitely many integers r with γ r+ > γ r and infinitely many integers r with γ r+ < γ r. Theorem 2 will be proved in Section 3. There we also argue that the smallest r satisfying γ r+ > γ r is probably larger than 0 25, and hence no amount of computer calculation today would detect this phenomenon. This behavior is closely linked to the classical problem of locating sign changes of π li, where π is the number of primes and dt ε dt li = 0 log t = lim + ε ε log t is Gauss approimation to π. Despite the oscillations, {γ r } can be shown on RH to approach e γ from above. If RH is false, {γ r } assumes values above and below e γ while converging to this value. THEOREM 3. Assume RH. Then γ r > e γ for all r 0. Moreover, we have 4 γ r = e γ gp r + pr r 2, where.95 g 2.05 for large. As we shall see later, lim sup g > 2 and lim inf g < 2. THEOREM 4. Assume RH is false. Then γ r < e γ for infinitely many r. In particular, Γ < e γ.

4 4 HAROLD G. DIAMOND AND KEVIN FORD COROLLARY. The Riemann Hypothesis is equivalent to the statement γ r > e γ for all r 0. It is relatively easy to find reasonable, unconditional lower bounds on Γ by making use of Theorem, Proposition and eplicit bounds for counting functions of primes. By Theorem 7 of [0], we have > e γ p log 2 log Theorem 6 of [0] states that p p p > log γ p pp 2 log >. Using Proposition and writing p = p + pp, we obtain for = p r 285 the bound In the last step we used γ r e γ log e γ log p + 2 log 2 2 log 2 γ + p p + p pp p + log + + log 2 log 2. pp < + log t + + log t 2 dt = 2. pp By the aforementioned computer calculations, γ r > e γ when p r < 0 9, and for p r > 0 9 the bound given above implies that γ r Therefore, we have unconditionally Γ Better lower bounds can be achieved by utilizing longer computer calculations, better bounds for prime counts [9], and some of the results from 4 below, especially An etremal property of {γ r } In this section we prove Theorem. Starting with an arbitrary finite set P of primes, we perform a sequence of operations on P, at each step either removing the largest prime from our set or replacing the largest prime with a smaller one. We stop when the resulting set is the first r primes, with 0 r #P. We make strategic choices of the operations to create a sequence of sets of primes P 0 = P, P,..., P k, where γp 0 > γp > > γp k with P k = {p, p 2,..., p r }, the first r primes. The method is simple to describe. Let Q = P j, which is not equal to any set {p, p 2,..., p s }, and with largest element t. Let Q = Q\{t}. If γq < γq, we set P j+ = Q. Otherwise, we set P j+ = Q {u}, where u is the smallest prime not in Q. We have u < t by assumption. It remains to show in the latter case that 2 γp j+ < γq. By 2, for any prime v Q, 2 2 γq {v} = γq v + log v =:γq fv, va A:= γ + p Q p. Observe that fv is strictly increasing for v < e A+ and strictly decreasing for v > e A+, and lim v fv =.

5 Generalized Euler constants 5 Thus fv > for e A+ v <. Since γq = γq ft γq, we have ft. It follows that u < t e A+ and hence fu < ft. Another application of 2 2, this time with v = u, proves 2 and the theorem follows. Define 3. The γ r s are not monotone 3 A:= γ + p By 2 2, we have thus p. γ r+ = γ r + r+, p r+ p r+ Ap r 3 2 γ r+ γ r Ap r r+. THEOREM 5. We have A log = Ω ± /2 log log log. Proof. First introduce 3 3 := p p n Λn n and θ:= p. Then 3 4 = p α = p p α p α α log / + p α p α = p p p log / 0. Since p log / /p, we have 3 5 = <p pp + Aside from an error of O, the first sum is p 2 = θ + p> /3 <p /2 p 2 p + O. p /3 2θt t 3 dt = /2 + O /2 log 3, using the bound θ log 3 which follows from the prime number theorem with a suitable error term. The second sum and error term in 3 5 are each O 2/3, and we deduce that 3 6 = + O. log 3 REMARK. Assuming RH and using the von Koch bound θ log 2, we obtain the sharper estimate = /2 + O 2/3. By 3 6, 3 7 A = γ + n Λn n + + O log 3.

6 6 HAROLD G. DIAMOND AND KEVIN FORD To analyze the above sum, introduce 3 8 R:= n For Rs > 0, we compute the Mellin transform 3 9 Λn n s R d = s log + γ. ζ ζ s + s 2 + γ s. The largest real singularity of the function on the right comes from the trivial zero of ζs + at s = 3. Let = β + iτ represent a generic nontrivial zero of ζs we avoid use of γ for I for obvious reasons. If RH is false, there is a zero β + iτ of ζ with β > /2, and a straightforward application of Landau s Oscillation Theorem [], Theorem 6.3 gives R = Ω ± β ε for every ε > 0. In this case, A log = Ω ± β ε, which is stronger than the assertion of the theorem. If RH is true, we may analyze R via the eplicit formula + 2n + 2n, 3 0 R 0 := 2 {R + R + } = n= where means lim T T. Equation 3 0 is deduced in a standard way from 3 9 by contour integration, and lim T T converges boundedly for in any fied compact set contained in,. cf. [3], Ch. 7, where a similar formula is given for ψ 0, as we now describe. In showing that ψ:= n Λn = + Ω ± /2 log log log, Littlewood [6] also cf. [3], Ch. 7 used the analogous eplicit formula and proved that ψ 0 := 2 {ψ+ + ψ } = ζ ζ 0 = Ω ± log log log. + Forming a difference of normalized sums over the non-trivial zeros, we obtain /2 /2. Thus and hence R = Ω ± /2 log log log. Therefore, in both cases RH false, RH true, we have and the theorem follows from 3 7 and 3 8. = /2 Ω ± log log log, R = Ω ± /2 log log log, n= 2n 2n By Theorem 5, there are arbitrarily large values of for which A < log. If p r is the largest prime, then Ap r = A < log < r+

7 Generalized Euler constants 7 for such. This implies by 3 2 that γ r+ > γ r. For the second part of Theorem 2, take large and satisfying A > log + /2 and let p r+ be the largest prime. By Bertrand s postulate, p r+ /2. Hence which implies γ r+ < γ r. Ap r = Ap r+ r+ p r+ log + /2 log /2 > log r+, Computations with PARI/GP reveal that γ r+ < γ r for all r with p r < 0 9. By 3 7, to find r such that γ r+ > γ r, we need to search for values of essentially satisfying R < /2. By 3 0, this boils down to finding values of u = log such that e iuτ iτ /2 >. =β+iτ Of course, the smallest zeros of ζs make the greatest contributions to this sum. Let lu be the truncated version of the preceding sum taken over the zeros with I T 0 := approimately 2 million zeros with positive imaginary part, together with their conjugates. A table of these zeros, accurate to within 3 0 9, is provided on Andrew Odlyzko s web page odlyzko/zeta tables/inde.html. In computations of lu, the errors in the values of the zeros contribute a total error of at most u u. I T 0 Computation using u-values at increments of 0 5 and an early abort strategy for u s having too small a sum over the first 000 zeros, indicates that lu 0.92 for 0 u Thus, it seems likely that the first r with γ r+ > γ r occurs when p r is of size at least e There is a possibility that the first occurence of γ r+ > γ r happens nearby, as l > Going out further, we find that l >.05, and an averaging method of R. S. Lehman [5] can be used to prove that γ r+ > γ r for many values of r in the vicinity of e Incidentally, for the problem of locating sign changes of π li, one must find values of u for which essentially e iuτ iτ + /2 <. =β+iτ A similarly truncated sum over zeros with I 600, 000 first attains values less than for positive u values when u [2]. 4. Proof of Theorem 3 Showing that γ r > e γ for all r 0 under RH requires eplicit estimates for prime numbers. Although sharper estimates are known cf. [9], older results of Rosser and Schoenfeld suffice for our purposes. The net lemma follows from Theorems 9 and 0 of [0]. LEMMA 4. We have θ.07 for > 0 and θ for 000. The preceeding lemma is unconditional. On RH, we can do better for large, such as the following results of Schoenfeld [], Theorem 0 and Corollary 2. LEMMA 4 2. Assume RH. Then log 2 θ < 8π 599

8 8 HAROLD G. DIAMOND AND KEVIN FORD and Mertens formula in the form and a familiar small calculation give It follows that 4 R 3 log2 + 6 log + 2 8π 8.4. log /p = log log + γ + o p log /p p n n n Λn n log n = p p a > Λn = log log + γ + o. n log n ap a = O = o. log We can obtain an eact epression for the last sum in terms of R defined in 3 8 by integrating by parts: Λn n log n = dt/t + drt 2 log t where = log log log log 2 + R log R2 log 2 + Rt dt 2 t log 2 t = log log + c + R log Rt dt t log 2 t, c:= 2 Rt dt t log 2 t R2 log log 2 = γ, log 2 by reference to 4 and the relation R = o. Thus Λn R 4 2 = log log + γ + n log n log n Let H denote the integral in 4 2 and define 4 3 := p p a > ap a. Rt t log 2 t dt. Using 2, engaging nearly all the preceding notation and writing p r =, we have { 4 4 γ r = e γ log ep R } log + H + R +. log We use Lemmas 4 and 4 2 to obtain eplicit estimates for H,, and. We shall show below that log 0. It is crucial for our arguments that this difference be small. Also, although one may use Lemma 4 2 to bound H, we shall obtain a much better inequality by using the eplicit formula 3 0 for R 0 which agrees with R a.e.. LEMMA 4 3. Assume RH. Then H log log 00.

9 Generalized Euler constants 9 Proof. Since R = o by the prime number theorem, we see that the integral defining H converges absolutely. We write X Rt H = lim X t log 2 t dt, and treat the integral for H as a finite integral in justifying term-wise operations. We now apply the eplicit formula 3 0 for R. For t 00, t 2n 2n t 3 and thus 4 5 t log 2 t n= n= t 2n 0.2 dt < 2n + 3 log /2 log 2. The series over zeta zeros in 3 0 converges boundedly to R 0 as T for in a compact region; by the preceding remark on the integral defining H, we can integrate the series term-wise. For each nontrivial zero, integration by parts gives and thus t 2 log 2 t dt = log t 2 log 2 t dt /2 log log 3 /2 log log t 2 log 3 t dt t 3/2 dt Since RH is assumed true, we have by [3], Ch. 2, 0 and, 2 = 2 = 2 R = 2 + γ log 4π = Putting these pieces together, we conclude that H log 2 t 2 log 2 t dt log log Under assumption of RH, we have { 4 6 H = /2 log 2 iτ 2 + 4ϑ log } ϑ, 5/2 where ϑ and ϑ. The series are each absolutely summable, and so the first series is an almost periodic function of log. Thus the values this series assumes are nearly repeated infinitely often. The other two terms in 4 6 converge to 0 as. Also, the mean value of H /2 log 2 is 0 integrate the first series; thus the first series in 4 6 assumes both positive and negative values. The lim sup and lim inf of H /2 log 2 are equal to the

10 0 HAROLD G. DIAMOND AND KEVIN FORD lim sup and lim inf of the first series in 4 6, and we have H = Ω ± /2 log 2. If one assumes that the zeros in the upper half-plane have imaginary parts which are linearly independent over the rationals unproved even under RH, but widely believed, then Kronecker s theorem implies that lim sup H log 2 = 2 + γ log 4π, lim inf H log 2 = 2 + γ log 4π. Continuing to assume RH but making no linear independence assumption on the τ s, we can show that lim inf H log < , 2 which is close to 2 + γ log 4π. Indeed, for =, the first series in 4 6 equals 2, and by almost periodicity this value is nearly repeated infinitely often. Also, so that = 4, 2 = 2 + /2 4. The net two lemmas are unconditional; i.e. they do not depend on RH. We do not try to obtain the sharpest estimates here. LEMMA 4 4. We have Proof. Using 3 4 and the upper bound for θ given in Lemma 4, p> p 2 θ + 2 p 2θt + t 3 dt + 2θ /2 < 3.05 / LEMMA 4 5. We have / log > log 2 = log 2 + O log Proof. By 3 4, we have 4 0 log.23 log 2 log = p a> log 0 6. p a log. a

11 Generalized Euler constants Each summand on the right side is clearly positive, proving the first part of the lemma. As shown in the proof of 3 5, the summands of associated with eponents a 3 make a total contribution of O 2/3. Thus the corresponding summands in 4 0 contribute O 2/3 / log. We handle the remaining term by partial summation, writing 4 <p p 2 log 2 = { t 2 log } dθt 2 log t = θ 2 2 log + θtt 3{ 2 log log t } 2 log 2 dt. t Using the prime number theorem with an error term θt t t log 2 t, the left side of 4 is seen to be { 2 = O log 3 + t 2 log t 2 log t } t 2 log 2 + t 2t 2 log 2 dt t = O log 3 + 2t 2 log 2 t dt 2 = log 2 + O log 3, proving the second part of the lemma. The proof of 4 7 shows the epression in 4 is a valid lower bound for / log. Inserting the estimates from Lemma 4 and applying integration by parts gives, for 0 6, log log = log Another application of integration by parts yields Finally, 2/ log / log t t 2 dt.07 2 dt t 2 log 2 t. dt t 2 log 2 t = 4 log 2 log 2 2dt t 2 log 3 t 4 log 2 log 2 6 log log 2. log 06 log 000 log 2. Combining the estimates, we obtain the third part of the lemma. dt t 2 log 2 t We are now set to complete the proof of Theorem 3. A short calculation using PARI/GP verifies that γ r > e γ for p r < 0 6. Assume now that = p r 0 6. By 4 4, { } { } 4 2 γ r = e γ R + R + + ep ep log log log + H.

12 2 HAROLD G. DIAMOND AND KEVIN FORD By Lemmas 4 2 and 4 4, R + log 3 log2 + 6 log π 8π log log By Taylor s theorem applied to y + log + y, if y then e y + y e 0.50y2. This, together with Lemmas 4 3 and 4 5, yields { γ r e γ ep log2 +.7 } log 2. Since /2 log 4 is decreasing for e 8, We conclude that log 2 log { } γ r e γ ep log 2 log log , which completes the proof of the first assertion. By combining 4 2 with Lemma 4 2, Lemma 4 4 and 4 8, we have { } γ r = e γ 2 ep H + log 2 + O log 3. Lemma 4 3 implies that H log 2 for large, and this proves 4. By the commentary following the proof of Lemma 4 3, we see that lim inf g < 2 and lim sup g > Analysis of γ r if RH is false Start with 4 2 and note that e y + y. Inserting the estimates from Lemma 4 5 gives { 5 γ r e γ } ep H + O log 2. Our goal is to show that H has large oscillations. Basically, a zero of ζs with real part β > /2 induces oscillations in H of size β ε, which will overwhelm the error term in 5. The Mellin transform of H does not eist because of the blow-up of the integrand near = ; however the function H log is bounded near =. LEMMA 5. For Rs > 0, we have s H log d = s 2 log s ζs + s + where Gs is a function that is analytic for Rs >. γ + Gs, s Proof. By 4 2, H log = log n Λn + R + log log log + γ. n log n

13 The Mellin transform of the sum is s log ζs + ; hence Let We have cf. 6.7 of [] s log n Generalized Euler constants 3 Λn n log n d = d log ζs + log ζs + = ds s s 2 ζ s +. s ζ f:= t t log t dt. f log = log log + γ log + O >, and note that a piecewise continuous function which is O/ has a Mellin transform which is analytic for Rs >. Also, s s f d = s f s s + d = d = log. log s Thus, s f log d = d s + ds s log s Recalling 3 9, the proof is complete. = s + s 2 log + s s 2 s + s +. We see that the Mellin transform of H log has no real singularities in the region Rs >. If ζs has a zero with real part β > /2, Landau s oscillation theorem implies that H log = Ω ± β ε for every ε > 0. Inequality 5 then implies that γ r < e γ for infinitely many r, proving Theorem 4. REMARK 2. We leave as an open problem to show that γ r > e γ for infinitely many r in case RH is false. If the supremum σ of real parts of zeros of ζs is strictly less than, then Landau s oscillation theorem immediately gives for every ε > 0, while a simpler argument shows that By 4 4, we have H = Ω ± σ ε R = O σ +ε. { R γ r = e γ 2 } ep H + O log 2 + O log and the desired result follows immediately. If ζs has a sequence of zeros with real parts approaching, Landau s theorem is too crude to show that H has larger oscillations than does R 2 / log 2. In this case, techniques of Pintz [7], [8] are perhaps useful. Acknowledement. The authors thank the referee for a careful reading of the paper and for helpful suggestions regarding the eposition. REFERENCES [] P. T. Bateman and H. G. Diamond, Analytic Number Theory, World Scientific, [2] C. Bays and R. H. Hudson, A new bound for the smallest with π > li, Math. Comp , [3] H. Davenport, Multiplicative number theory, 3rd ed. Revised and with a preface by H. L. Montgomery. Graduate Tets in Mathematics, 74. Springer-Verlag, New York, [4] J. Havil, Gamma : Eploring Euler s constant; With a foreword by Freeman Dyson, Princeton University Press, Princeton, NJ, [5] R. S. Lehman, On the difference π li, Acta Arith. 966,

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