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Please output this exact text, with no changes at all except for fixing the misspellings. Please leave all other stylistic decisions like commas and US vs British spellings as in the original text.

Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality forhten logarithm ofhten modulus of the Riemann zeta-fucntion on the critical line in terms of a Dirichlet polynomial over primes adn prime powers. Our proof useshten Guinand-Weil explicit formula in conjunction witht extremal one-sided bandlimited approximations forhten Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement ofther sharpest known upper bound (due to Chandee and Soundararajan) for the modulus ofhten zeta funtion onhten critical line assuming RH, by providing explicit lower-order terms.

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金声玉亮2.0 Inspired by a result of Soundararajan, assuming the Riemann Hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
deepseek-r1

Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with the extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.		 Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
gpto1 Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
gpt-4o-2024-08-06 Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
DeepSeek-Coder-V2-Lite-Instruct Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
Meta-Llama-3.1-405B-Instruct-Turbo Here is the text with the misspellings corrected:

Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.		 Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
Meta-Llama-3.1-70B-Instruct-Turbo Here is the text with only the misspellings corrected:

Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.		 Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
Qwen2-72B-Instruct Inspired by a result of Soundararajan, assuming the Riemann Hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
baichuan4 Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality forhten logarithm ofhten modulus of the Riemann zeta-fucntion on the critical line in terms of a Dirichlet polynomial over primes adn prime powers. Our proof useshten Guinand-Weil explicit formula in conjunction witht extremal one-sided bandlimited approximations forhten Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement ofther sharpest known upper bound (due to Chandee and Soundararajan) for the modulus ofhten zeta funtion onhten critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
doubao Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
gpto1 Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
kimi Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guivarc'h-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
minimax Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
wenxinyiyan Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
zhipu Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.
deepseek-v3 Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.