On Lerch's formula and zeros of the quadrilateral zeta function

2020/01/07 by Nakamura, Takashi
#FOS: Mathematics #Number Theory (math.NT) #Primary 11M35 #Secondary 11M26

paper · doi:10.48550/arxiv.2001.01981

Abstract

Let 0 < a ≤ 1/2 and define the quadrilateral zeta function by 2Q(s,a) := ζ(s,a) + ζ(s,1-a) + \rmLi_s (e^2πia) + \rmLi_s(e^2πi(1-a)), where ζ(s,a) is the Hurwitz zeta function and \rmLi_s (e^2πia) is the periodic zeta function. In the present paper, we show that there exists a unique real number a₀ ∈ (0,1/2) such that Q(σ, a₀) has a unique double real zero at σ= 1/2 when σ∈ (0,1), for any a ∈ (a₀,1/2], the function Q(σ, a) has no zero in the open interval σ∈ (0,1) and for any a ∈ (0,a₀), the function Q(σ, a) has at least two real zeros in σ∈ (0,1). Moreover, we prove that Q(s,a) has infinitely many complex zeros in the region of absolute convergence and the critical strip when a ∈ ℚ ∩ (0,1/2) ∖ \1/6, 1/4, 1/3\. The Lerch formula, Hadamard product formula, Riemann-von Mangoldt formula for Q(s,a) are also shown.

Related

• Zeros and the functional equation of the quadrilateral zeta function
• Functional equation and zeros on the critical line of the quadrilateral zeta function
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• Real zeros of Hurwitz-Lerch zeta functions in the interval (-1,0)
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