On the zeros of the Lerch zeta-function. III

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1 On the zeros of the Lerch zeta-function. III Ramūnas GARUNKŠIS (Vilnius University) Abstract. We investigate uper bounds for the number of zeros of the Lerch zeta function. Keywords: Lerch zeta-function, zeros of the Lerch zeta-function.. Introduction Let s = σ + it be a complex variable. he Lerch zeta-function L(λ,, s), for σ >, is defined by the following Dirichlet series L(λ,, s) = m= (m + ) s, where λ, are real numbers, <, and by analytic continuation othervise (see [5], [6]). Further we suppose that < λ <. In [4] A. Laurinčikas for the function Z(s, λ) = m s = e 2πiλ L(λ,, s), σ >, λ = a/q, (a, q) =, < a < q, obtained the following zero-distribution rezults. heorem A. Suppose that q is a prime number. hen there exists a constant c = c(λ) such that for sufficiently large the function Z(s, λ) has more than c zeros in the region σ >, t <. heorem B. Suppose there exist at least two primitive characters modulo q. hen for any σ, σ 2, /2 < σ < σ 2 <, there exists a constant c = c(λ, σ, σ 2 ) > such that for sufficiently large the function Z(s, λ) has more than c zeros in the region σ < σ < σ 2, t <. By A (λ, ; a, b) we will denote the following assertion: For any σ, σ 2, a < σ < σ 2 < b, there exists a constant c = c(λ,, σ, σ 2 ) > such that for sufficiently large the function L(λ,, s) has more than c zeros in the rectangle σ < σ < σ 2, t <. Partially supported by Lithuanian State Studies and Science Foundation.

2 In [] for the Lerch zeta function were obtained the following results. heorem C. Let be a nonrational number. hen there exists δ = δ(λ, ), < δ <, such that the assertion A (λ, ;, + δ) is true. If is a transcendental number, then we can take δ =.6. heorem D. Let be a transcendental number. hen the assertion A (λ, ; /2, ) is true. Let N(λ,, σ, ) denote the number of zeros of L(λ,, s) in the region {s Re s > σ, < Im s }. In [2] it is proved that L(λ,, s), for σ +. () In this note we investigate the uper bounds for the number of zeros of the Lerch zeta function. Let B η denote a number bounded by a constant depending on η. heorem. Let /2 +, then + N(λ, ; σ, ) d σ = log + log L(λ,, σ + it) d t + B log,. By ζ(s, ) we denote the Hurwitz zeta-function, i.e. where <. ζ(s, ) = m= (m + ) s, σ >, heorem 2. Let σ > /2, then for any fixed /2 < σ < σ and, we have ζ(2σ N(λ, ; σ, ) log(2σ, )) + R(σ, ), 2(σ σ ) where R(σ, ) = { Bσ 2 2σ for 2 < σ <, B σ log for σ. 2. Lemmas We will use a lemma of Littlewood [7, 9.9]. Let ϕ(s) be a meromorphic function on the rectangle D with vertices +i, β +i, β +i, +i, and let ϕ(s) be regular and nonvanishing on the line σ = β. hen ϕ(s) is regular in some neighbourhood of the line = β. In this neighbourhood we define a function F(s) = log ϕ(s), by choosing some 2

3 branch of the log ϕ(s). On other points of the rectangle we define F(s) by continuation of the log(β + it) left from β + it to σ + it. If a zero is reached, we use F(s) = lim F(σ + it + iε). ε + Let ν(σ, ) mean a difference between a number of zeros and a number of poles of ϕ(s) in a rectangle σ < σ β, < t. hen we have the following lemma. Lemma. F(s) d s = 2πi β ν(σ, ) d σ where the integral on the left we take around the contour of D. Lemma 2. For any, σ >, we have where k = k( ). Lemma is proved in [2]. L(λ,, s) = B λ t k. Lemma 3. Let σ > /2. hen, for, L(λ,, s) 2 d t = ζ(2σ, ) + r(σ, ), where B σ 2 2σ, for 2 < σ <, r(σ, ) = B σ log, for σ =, B σ, for σ >. For the proof see [4]. 3. Proofs of theorems Proof of heorem. Let σ > + be arbitrary large number. hen from Lemma and () we have for /2 + that 2π + σ + N(σ, ; λ, ) d σ = log L(λ,, + it) d t argl(σ + i) d σ + K(, σ ) = I + I 2 + I 3 + K(, σ ) 3 log L(λ,, σ + it) d t

4 where K(, σ ) does not depend on. First we evaluate the integral I 2. I 2 = log ( σ it + = σ log + ( m+ )σ+it log + ( m+ ) d t )σ+it d t = I 2 + I 22. (2) It is clear that there exists σ > +, such that a modulo of the sum in I 22 is less than for σ > σ and t R. For such σ by the Maclaurin formula we obtain I 22 = = Re ( [ ( ) n ( Re n n= ( ) n n= ( n m = m 2= ( m+ )σ+it... ) n ]) d t e 2πiλ(m+m2+...+mn) ( (m+)(m 2+)...(m n+) ) σ m n= n ) it d t = B (m + )(m 2 + )...(m n + ) We can choose σ big enough such that ( ) σ <. m + n= ( ( ) ) σ n. n m + (3) From this and (3) we have that and from (2) I 22 = B, I 2 = σ log + B. Now it remains to estimate the I 3. We define the function Φ(s) = e i log L(λ,, s), hen I 3 = σ arg Φ(σ + it) d σ (σ ) log. (4) 4

5 It is easily seen that the leading term of the Dirichlet series for Φ(s) is positive at s = σ + i. Denote by q the number of zeros of Re Φ(s) on the interval J = ( + i, σ + i), and divide J into at most q + subintervals in each of which Re Φ(s) is of constant sign. hen the variation of argφ(s) does not exceed π in each subinterval, and we obtain argφ(s) σ+i σ +i (q + )π. (5) o estimate q we set f(z) = 2( Φ(z + i) + Φ(z + i) ). First we note that f(z) is an entire function, and if z = σ is real, then f(σ) = Re Φ(σ + i). (6) Let n( ) stand for the number of zeros of f(z) in the disc z σ, and let r = 2(σ ), r = r/2. hen, clearly, r n( ) r d n(r ) r d s = n(r )log 2, and the well-known Jensen theorem yield n(r ) 2π log 2 2π log f(re iθ + σ ) d θ log 2 log f(σ ). (7) By (6) ( f(σ ) = Re + σ σ ( +m )σ+i ) σ σ ( +m )σ. For sufficiently large σ this is /(2) σ, say. Hence and from (7), using Lemma 2, we obtain that n(r ) = B log. (8) By (6), the number of zeros of ReΦ(s) on J is equal to the same number of zeros of f(z) on (, σ ). By the definition (, σ ) is contained in the disc z σ r. his, (8), (5) and (4) show that I 3 = (σ ) log + B log. 5

6 he theorem is proved for is not an ordinate of the zero of the L(λ,, s). For others theorem is true in view of a continuity. Proof of heorem 2. Using the concavity of the logarithm, from Lemma 3, we have log L(λ,, σ + it) d t = log L(λ,, σ + it) 2 d t 2 ( 2 log L(λ,, σ + it) ) 2 d t = 2 ( log ζ(2σ, ) + hen in view of heorem we obtain r(σ, ) ). σ + N(λ, ; σ, ) N(λ, ; σ, ) d σ N(λ, ; σ, ) d σ σ σ σ σ σ σ ( = σ σ 2 log ( ζ(2σ, ) + r(σ ), ) ) + σ log + B log. From this the theorem follows. References. R. Garunkštis, On zeros of the Lerch zeta-function. II. Preprint R. Garunkštis and A. Laurinčikas, On zeros of the Lerch zeta function, in: Proceedings of the Conf. Number heory and its Applications, Kyoto, 997 (to appear). 3. A. Laurinčikas, On zeros of some Dirichlet series, Liet. Matem. Rink., 24(4 (984), 6-26 (in Russian). 4. A. Laurinčikas, On the mean square of the Lerch zeta function, LMD mokslo darbai: Specialus Liet. Matem. Rink. priedas, III tomas, (spaudoje). 5. M. Lerch, Note sur la fonction K(w, x, s) = n exp{2πix}(n w) s, Acta Math., (887), M. Mikolás, New proof and extention of the functional equality of Lerch s zeta-function, Ann. Univ. Sci. Budapest. Sec. Math., 4 (97), E. C. itchmarsh, he heory of the Riemann Zeta-Function, Oxford, 95. Apie Lercho dzeta funkcijos nulius. III R. Garunkštis Straipsnyje nagrinėjami Lercho dzeta funkcijos nuliu skaičiaus i verčiai iš viršaus. 6